Preparation for the exam physics molecular physics. Molecular physics

We continue to analyze tasks from the first part of the Unified State Exam in Physics, dedicated to the topic “Molecular Physics and Thermodynamics”. As usual, all solutions are provided with detailed comments from a physics tutor. There is also a video analysis of all the proposed tasks. At the end of the article you can find links to analyzes of other tasks from the Unified State Exam in physics.

Thermodynamic equilibrium is understood as the state of a system in which its macroscopic parameters do not change over time. This state will be achieved when the temperatures of nitrogen and oxygen in the vessel are equalized. All other parameters will depend on the mass of each gas and, in general, will not be the same, even when thermodynamic equilibrium occurs. Correct answer: 1.

In an isobaric process, the volume V and temperature T

So, addiction V from T should be directly proportional, and if the temperature decreases, then the volume should decrease. Schedule 4 is suitable.

The efficiency of a heat engine is determined by the formula:

Here A- work done per cycle, Q 1 is the amount of heat received by the working fluid per cycle from the heater. Calculations give the following result: kJ.

11. When studying isoprocesses, a closed vessel of variable volume was used, filled with air and connected to a pressure gauge. The volume of the vessel is slowly increased, keeping the air pressure in it constant. How do the temperature of the air in the vessel and its density change? For each quantity, determine the corresponding nature of its change:

1) will increase

2) will decrease

3) will not change

Write down the selected numbers for each physical quantity in the table. The numbers in the answer may be repeated.

The process is isobaric. In an isobaric process, the volume V and temperature T ideal gas are related by the relation:

So, addiction V from T directly proportional, that is, as the volume increases, the temperature also increases.

The density of a substance is related to mass m and volume V ratio:

So, at constant mass m addiction ρ from V inversely proportional, that is, if the volume increases, then the density decreases.

Correct answer: 12.

12. The figure shows a diagram of four successive changes in the state of 2 moles of an ideal gas. In which process is the work of the gas positive and minimal in value, and in which process is the work of external forces positive and minimal in value? Match these processes with the process numbers on the diagram.
For each position in the first column, select the corresponding position from the second column and write down the selected numbers in the table under the corresponding letters.

The work of a gas is numerically equal to the area under the graph of the gas process in coordinates. By sign, it is positive in a process that occurs with an increase in volume, and negative in the opposite case. The work of external forces, in turn, is equal in magnitude and opposite in sign to the work of the gas in the same process.

That is, the work of the gas is positive in processes 1 and 2. Moreover, in process 2 it is less than in process 1, since the area of the yellow trapezoid in the figure is less than the area of the brown trapezoid:

On the contrary, the work of gas is negative in processes 3 and 4, which means that in these processes the work of external forces is positive. Moreover, in process 4 it is less than in process 3, since the area of the blue trapezoid in the figure is less than the area of the red trapezoid:

So the correct answer is: 42.

This was the last assignment on the topic “Molecular Physics and Thermodynamics” from the first part of the Unified State Exam in Physics. Look for analysis of tasks on mechanics.

Material prepared by Sergei Valerievich

Molecular kinetic theory called the doctrine of the structure and properties of matter based on the idea of the existence of atoms and molecules as the smallest particles of a chemical substance. The molecular kinetic theory is based on three main principles:

All substances - liquid, solid and gaseous - are formed from tiny particles - molecules, which themselves consist of atoms(“elementary molecules”). The molecules of a chemical substance can be simple or complex and consist of one or more atoms. Molecules and atoms are electrically neutral particles. Under certain conditions, molecules and atoms can acquire additional electrical charge and turn into positive or negative ions (anions and cations, respectively).
Atoms and molecules are in continuous chaotic movement and interaction, the speed of which depends on temperature, and the nature of which depends on the state of aggregation of the substance.
Particles interact with each other by forces that are electrical in nature. The gravitational interaction between particles is negligible.

Atom– the smallest chemically indivisible particle of an element (iron, helium, oxygen atom). Molecule- the smallest particle of a substance that retains its chemical properties. The molecule consists of one or more atoms (water - H 2 O - 1 oxygen atom and 2 hydrogen atoms). And he– an atom or molecule that has one or more electrons extra (or electrons missing).

Molecules are extremely small in size. Simple monatomic molecules have a size of the order of 10–10 m. Complex polyatomic molecules can have sizes hundreds and thousands of times larger.

The random chaotic movement of molecules is called thermal motion. The kinetic energy of thermal motion increases with increasing temperature. At low temperatures, molecules condense into a liquid or solid. As the temperature increases, the average kinetic energy of a molecule becomes greater, the molecules fly apart, and a gaseous substance is formed.

In solids, molecules undergo random vibrations around fixed centers (equilibrium positions). These centers can be located in space in an irregular manner (amorphous bodies) or form ordered volumetric structures (crystalline bodies).

In liquids, molecules have much greater freedom for thermal movement. They are not tied to specific centers and can move throughout the entire volume of liquid. This explains the fluidity of liquids.

In gases, the distances between molecules are usually much larger than their sizes. The forces of interaction between molecules at such large distances are small, and each molecule moves along a straight line until the next collision with another molecule or with the wall of the container. The average distance between air molecules under normal conditions is about 10 –8 m, that is, hundreds of times greater than the size of the molecules. The weak interaction between molecules explains the ability of gases to expand and fill the entire volume of the vessel. In the limit, when the interaction tends to zero, we arrive at the idea of an ideal gas.

Ideal gas is a gas whose molecules do not interact with each other, with the exception of elastic collision processes, and are considered material points.

In molecular kinetic theory, the amount of matter is considered to be proportional to the number of particles. The unit of quantity of a substance is called a mole (mole). Mole- this is the amount of substance containing the same number of particles (molecules) as there are atoms in 0.012 kg of carbon 12 C. A carbon molecule consists of one atom. Thus, one mole of any substance contains the same number of particles (molecules). This number is called Avogadro's constant: N A = 6.022·10 23 mol –1.

Avogadro's constant is one of the most important constants in molecular kinetic theory. Quantity of substance is defined as the ratio of the number N particles (molecules) of matter to Avogadro's constant N A, or as the ratio of mass to molar mass:

The mass of one mole of a substance is usually called molar mass M. Molar mass is equal to the product of mass m 0 of one molecule of a given substance per Avogadro constant (that is, per number of particles in one mole). Molar mass is expressed in kilograms per mole (kg/mol). For substances whose molecules consist of a single atom, the term atomic mass is often used. In the periodic table, molar mass is indicated in grams per mole. Thus we have another formula:

Where: M- molar mass, N A – Avogadro’s number, m 0 – mass of one particle of matter, N– the number of particles of a substance contained in the mass of a substance m. In addition, you will need the concept concentrations(number of particles per unit volume):

Let us also recall that the density, volume and mass of a body are related by the following formula:

If the problem involves a mixture of substances, then we talk about the average molar mass and the average density of the substance. As when calculating the average speed of uneven movement, these values are determined by the total masses of the mixture:

Do not forget that the total amount of a substance is always equal to the sum of the amounts of substances included in the mixture, and you need to be careful with the volume. Gas mixture volume Not equal to the sum of the volumes of gases included in the mixture. So, 1 cubic meter of air contains 1 cubic meter of oxygen, 1 cubic meter of nitrogen, 1 cubic meter of carbon dioxide, etc. For solids and liquids (unless otherwise specified in the condition), we can assume that the volume of the mixture is equal to the sum of the volumes of its parts.

Basic equation of MKT ideal gas

As they move, gas molecules continually collide with each other. Because of this, the characteristics of their movement change, therefore, when speaking about impulses, velocities, and kinetic energies of molecules, we always mean the average values of these quantities.

The number of collisions of gas molecules under normal conditions with other molecules is measured millions of times per second. If we neglect the size and interaction of molecules (as in the ideal gas model), then we can assume that between successive collisions the molecules move uniformly and rectilinearly. Naturally, when approaching the wall of the vessel in which the gas is located, the molecule also experiences a collision with the wall. All collisions of molecules with each other and with the walls of the container are considered absolutely elastic collisions of balls. When it collides with a wall, the momentum of the molecule changes, which means that a force acts on the molecule from the side of the wall (remember Newton’s second law). But according to Newton's third law, with exactly the same force directed in the opposite direction, the molecule acts on the wall, exerting pressure on it. The totality of all impacts of all molecules on the wall of the vessel leads to the appearance of gas pressure. Gas pressure is the result of collisions of molecules with the walls of the container. If there is no wall or any other obstacle for the molecules, then the very concept of pressure loses its meaning. For example, it is completely unscientific to talk about pressure in the center of the room, because there the molecules do not press on the wall. Why then, when we place a barometer there, are we surprised to find that it shows some kind of pressure? Right! Because the barometer itself is the very wall on which the molecules press.

Since pressure is a consequence of the impacts of molecules on the wall of a vessel, it is obvious that its value should depend on the characteristics of individual molecules (on the average characteristics, of course, you remember that the speeds of all molecules are different). This dependence is expressed the basic equation of the molecular kinetic theory of an ideal gas:

Where: p- gas pressure, n- concentration of its molecules, m 0 - mass of one molecule, v kv - root mean square speed (note that the equation itself contains the square of the root mean square speed). The physical meaning of this equation is that it establishes a connection between the characteristics of the entire gas (pressure) and the parameters of the movement of individual molecules, that is, the connection between the macro- and microworld.

Corollaries from the basic MKT equation

As already noted in the previous paragraph, the speed of thermal movement of molecules is determined by the temperature of the substance. For an ideal gas, this dependence is expressed by simple formulas for root mean square speed movement of gas molecules:

Where: k= 1.38∙10 –23 J/K – Boltzmann constant, T– absolute temperature. Let’s immediately make a reservation that in future in all problems you should, without hesitation, convert the temperature into kelvins from degrees Celsius (except for problems on the heat balance equation). Law of Three Constants:

Where: R= 8.31 J/(mol∙K) – universal gas constant. The next important formula is the formula for average kinetic energy of translational motion of gas molecules:

It turns out that the average kinetic energy of the translational motion of molecules depends only on temperature and is the same at a given temperature for all molecules. And finally, the most important and frequently used consequences from the basic MKT equation are the following formulas:

Temperature measurement

The concept of temperature is closely related to the concept of thermal equilibrium. Bodies in contact with each other can exchange energy. The energy transferred from one body to another during thermal contact is called the amount of heat.

Thermal equilibrium- this is a state of a system of bodies in thermal contact in which there is no heat transfer from one body to another, and all macroscopic parameters of the bodies remain unchanged. Temperature is a physical parameter that is the same for all bodies in thermal equilibrium.

To measure temperature, physical instruments are used - thermometers, in which the temperature value is judged by a change in any physical parameter. To create a thermometer, you must select a thermometric substance (for example, mercury, alcohol) and a thermometric quantity that characterizes the property of the substance (for example, the length of a mercury or alcohol column). Various thermometer designs use various physical properties of a substance (for example, a change in the linear dimensions of solids or a change in the electrical resistance of conductors when heated).

Thermometers must be calibrated. To do this, they are brought into thermal contact with bodies whose temperatures are considered given. Most often, simple natural systems are used in which the temperature remains unchanged despite heat exchange with the environment - a mixture of ice and water and a mixture of water and steam when boiling at normal atmospheric pressure. On the Celsius temperature scale, the melting point of ice is assigned a temperature of 0°C, and the boiling point of water: 100°C. The change in the length of the liquid column in the capillaries of the thermometer per one hundredth of the length between the marks of 0°C and 100°C is taken equal to 1°C.

The English physicist W. Kelvin (Thomson) in 1848 proposed using the point of zero gas pressure to construct a new temperature scale (Kelvin scale). In this scale, the temperature unit is the same as in the Celsius scale, but the zero point is shifted:

In this case, a temperature change of 1ºC corresponds to a temperature change of 1 K. Temperature changes on the Celsius and Kelvin scales are equal. In the SI system, the unit of temperature measured on the Kelvin scale is called kelvin and denoted by the letter K. For example, room temperature T C = 20°C on the Kelvin scale is T K = 293 K. The Kelvin temperature scale is called the absolute temperature scale. It turns out to be most convenient when constructing physical theories.

Equation of state of an ideal gas or Clapeyron-Mendeleev equation

Equation of state of an ideal gas is another consequence of the basic MKT equation and is written in the form:

This equation establishes a relationship between the main parameters of the state of an ideal gas: pressure, volume, amount of substance and temperature. It is very important that these parameters are interconnected; changing any of them will inevitably lead to changing at least one more. That is why this equation is called the equation of state of an ideal gas. It was discovered first for one mole of gas by Clapeyron, and subsequently generalized to the case of a larger number of moles by Mendeleev.

If the gas temperature is T n = 273 K (0°C), and pressure p n = 1 atm = 1 10 5 Pa, then they say that the gas is at normal conditions.

Gas laws

Solving problems for calculating gas parameters is greatly simplified if you know which law and which formula to apply. So, let's look at the basic gas laws.

1. Avogadro's law. One mole of any substance contains the same number of structural elements, equal to Avogadro's number.

2. Dalton's law. The pressure of a mixture of gases is equal to the sum of the partial pressures of the gases included in this mixture:

The partial pressure of a gas is the pressure it would produce if all the other gases suddenly disappeared from the mixture. For example, air pressure is equal to the sum of the partial pressures of nitrogen, oxygen, carbon dioxide and other impurities. In this case, each of the gases in the mixture occupies the entire volume provided to it, that is, the volume of each of the gases is equal to the volume of the mixture.

3. Boyle-Mariotte law. If the mass and temperature of the gas remain constant, then the product of the gas pressure and its volume does not change, therefore:

A process occurring at a constant temperature is called isothermal. Note that this simple form of the Boyle-Marriott law only holds if the mass of the gas remains constant.

4. Gay-Lussac's law. Gay-Lussac's law itself is not of particular value when preparing for exams, so we will give only a corollary from it. If the mass and pressure of the gas remain constant, then the ratio of the volume of the gas to its absolute temperature does not change, therefore:

A process that occurs at constant pressure is called isobaric or isobaric. Note that this simple form of Gay-Lussac's law only holds if the mass of the gas remains constant. Don't forget about converting temperature from degrees Celsius to Kelvin.

5. Charles's law. Like Gay-Lussac’s law, Charles’s law in its exact formulation is not important for us, so we will only give a corollary from it. If the mass and volume of the gas remain constant, then the ratio of the gas pressure to its absolute temperature does not change, therefore:

A process occurring at constant volume is called isochoric or isochoric. Note that this simple form of Charles's law only holds if the mass of the gas remains constant. Don't forget about converting temperature from degrees Celsius to Kelvin.

6. Universal gas law (Clapeyron). At a constant mass of a gas, the ratio of the product of its pressure and volume to temperature does not change, therefore:

Please note that the mass must remain the same, and do not forget about kelvins.

So, there are several gas laws. We list the signs that you need to use one of them when solving a problem:

Avogadro's law applies to all problems involving the number of molecules.
Dalton's law applies to all problems involving a mixture of gases.
Charles's law is used in problems where the volume of gas remains constant. Usually this is either stated explicitly, or the problem contains the words “gas in a closed vessel without a piston.”
Gay-Lussac's law is applied if the gas pressure remains unchanged. Look for the words “gas in a vessel closed by a movable piston” or “gas in an open vessel” in the problems. Sometimes nothing is said about the vessel, but according to the condition it is clear that it communicates with the atmosphere. Then it is assumed that atmospheric pressure always remains unchanged (unless otherwise stated in the condition).
Boyle-Marriott law. This is where it's most difficult. It’s good if the problem says that the temperature of the gas is constant. It’s a little worse if the word “slow” is present in the condition. For example, a gas is slowly compressed or slowly expanded. It is even worse if it is said that the gas is closed by a heat-non-conducting piston. Finally, it’s really bad if nothing is said about the temperature, but from the condition it can be assumed that it does not change. Usually in this case, students apply the Boyle-Marriott law out of despair.
Universal gas law. It is used if the mass of the gas is constant (for example, the gas is in a closed vessel), but according to the condition it is clear that all other parameters (pressure, volume, temperature) change. In general, you can often use the Clapeyron-Mendeleev equation instead of the universal law; you will get the correct answer, only you will write two extra letters in each formula.

Graphic representation of isoprocesses

In many branches of physics, it is convenient to depict the dependence of quantities on each other graphically. This makes it easier to understand the relationships between parameters occurring in a process system. This approach is very often used in molecular physics. The main parameters describing the state of an ideal gas are pressure, volume and temperature. The graphical method for solving problems consists of depicting the relationship of these parameters in various gas coordinates. There are three main types of gas coordinates: ( p; V), (p; T) And ( V; T). Note that these are just the basic (most common types of coordinates). The imagination of the writers of problems and tests is not limited, so you can come across any other coordinates. So, let us depict the main gas processes in the main gas coordinates.

Isobaric process (p = const)

An isobaric process is a process that occurs at constant pressure and mass of gas. As follows from the equation of state of an ideal gas, in this case the volume changes in direct proportion to the temperature. Graphs of the isobaric process in coordinates R–V; V–T And R–T have the following form:

V–T coordinates is directed exactly to the origin, but this graph can never start directly from the origin, since at very low temperatures gas turns into liquid and the dependence of volume on temperature changes.

Isochoric process (V = const)

An isochoric process is the process of heating or cooling a gas at a constant volume and provided that the amount of substance in the vessel remains unchanged. As follows from the equation of state of an ideal gas, under these conditions the gas pressure changes in direct proportion to its absolute temperature. Graphs of an isochoric process in coordinates R–V; R–T And V–T have the following form:

Please note that the continuation of the graph in p–T coordinates is directed exactly to the origin, but this graph can never start directly from the origin, since gas turns into liquid at very low temperatures.

Isothermal process (T = const)

An isothermal process is a process that occurs at a constant temperature. From the equation of state of an ideal gas it follows that at a constant temperature and a constant amount of substance in the vessel, the product of the gas pressure and its volume must remain constant. Graphs of an isothermal process in coordinates R–V; R–T And V–T have the following form:

Note that when performing tasks on graphs in molecular physics Not special accuracy is required in plotting coordinates along the corresponding axes (for example, so that the coordinates p 1 and p 2 two states of gas in the system p(V) coincided with the coordinates p 1 and p 2 of these states in the system p(T). Firstly, these are different coordinate systems in which different scales can be chosen, and secondly, this is an unnecessary mathematical formality that distracts from the main thing - the analysis of the physical situation. The main requirement: that the quality of the graphs be correct.

Nonisoprocesses

In problems of this type, all three main gas parameters change: pressure, volume and temperature. Only the mass of the gas remains constant. The simplest case is if the problem is solved “head-on” using the universal gas law. It’s a little more difficult if you need to find an equation for a process that describes a change in the state of a gas, or analyze the behavior of gas parameters using this equation. Then you need to act like this. Write down this equation of the process and the universal gas law (or the Clapeyron-Mendeleev equation, whichever is more convenient for you) and consistently eliminate unnecessary quantities from them.

Change in quantity or mass of a substance

In essence, there is nothing complicated in such tasks. You just need to remember that the gas laws are not satisfied, since the formulations of any of them say “at constant mass.” Therefore, we act simply. We write the Clapeyron-Mendeleev equation for the initial and final states of the gas and solve the problem.

Baffles or pistons

In problems of this type, gas laws are again applied, and the following remarks must be taken into account:

Firstly, gas does not pass through the partition, that is, the mass of gas in each part of the vessel remains unchanged, and thus the gas laws are satisfied for each part of the vessel.
Secondly, if the partition is heat-non-conducting, then when the gas is heated or cooled in one part of the vessel, the temperature of the gas in the second part will remain unchanged.
Thirdly, if the partition is movable, then the pressures on both sides are equal at any given moment in time (but this pressure, equal on both sides, can change over time).
And then we write gas laws for each gas separately and solve the problem.

Gas laws and hydrostatics

The specificity of the problems is that in the pressure it will be necessary to take into account the “add-on weights” associated with the pressure of the liquid column. What options might there be:

A container containing gas is submerged under water. The pressure in the vessel will be equal to: p = p atm + ρgh, Where: h– immersion depth.
Horizontal the tube is closed from the atmosphere by a column of mercury (or other liquid). The gas pressure in the tube is exactly equal to: p = p atm atmospheric, since a horizontal column of mercury does not exert pressure on the gas.
Vertical the gas tube is closed on top with a column of mercury (or other liquid). Gas pressure in the tube: p = p atm + ρgh, Where: h– height of the mercury column.
A vertical narrow tube containing gas is turned with the open end down and is sealed with a column of mercury (or other liquid). Gas pressure in the tube: p = p atm – ρgh, Where: h– height of the mercury column. The “–” sign is used because mercury does not compress, but stretches the gas. Students often ask why the mercury does not flow out of the tube. Indeed, if the tube were wide, the mercury would flow down the walls. And so, since the tube is very narrow, surface tension does not allow the mercury to rupture in the middle and let air in, and the gas pressure inside (less than atmospheric) keeps the mercury from flowing out.

Once you have been able to correctly record the gas pressure in the tube, apply one of the gas laws (usually Boyle-Mariotte, since most of these processes are isothermal, or the universal gas law). Apply the chosen law for gas (in no case for liquid) and solve the problem.

Thermal expansion of bodies

As the temperature rises, the intensity of the thermal movement of particles of a substance increases. This causes the molecules to more “actively” repel each other. Because of this, most bodies increase in size when heated. Don't make the typical mistake; atoms and molecules themselves do not expand when heated. Only the empty spaces between the molecules increase. The thermal expansion of gases is described by Gay-Lussac's law. The thermal expansion of liquids obeys the following law:

Where: V 0 – volume of liquid at 0°C, V- at a temperature t, γ – coefficient of volumetric expansion of the liquid. Please note that all temperatures in this topic must be taken in degrees Celsius. The coefficient of volumetric expansion depends on the type of liquid (and on temperature, which is not taken into account in most problems). Please note that the numerical value of the coefficient, expressed in 1/°C or 1/K, is the same, since heating a body by 1°C is the same as heating it by 1 K (and not by 274 K).

For expansion of solids Three formulas are used to describe the change in linear dimensions, area and volume of a body:

Where: l 0 , S 0 , V 0 – length, surface area and volume of the body at 0°C, respectively, α – coefficient of linear expansion of the body. The coefficient of linear expansion depends on the type of body (and on temperature, which is not taken into account in most problems) and is measured in 1/°C or 1/K.

Learn all the formulas and laws in physics, and formulas and methods in mathematics. In fact, this is also very simple to do; there are only about 200 necessary formulas in physics, and even a little less in mathematics. In each of these subjects there are about a dozen standard methods for solving problems of a basic level of complexity, which can also be learned, and thus, completely automatically and without difficulty solving most of the CT at the right time. After this, you will only have to think about the most difficult tasks.

Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to decide on both options. Again, on the CT, in addition to the ability to quickly and efficiently solve problems, and knowledge of formulas and methods, you must also be able to properly plan time, distribute forces, and most importantly, correctly fill out the answer form, without confusing the numbers of answers and problems, or your own last name. Also, during RT, it is important to get used to the style of asking questions in problems, which may seem very unusual to an unprepared person at the DT.

Successful, diligent and responsible implementation of these three points will allow you to show an excellent result at the CT, the maximum of what you are capable of.

Found a mistake?

If you think you have found an error in the training materials, please write about it by email. You can also report an error on the social network (). In the letter, indicate the subject (physics or mathematics), the name or number of the topic or test, the number of the problem, or the place in the text (page) where, in your opinion, there is an error. Also describe what the suspected error is. Your letter will not go unnoticed, the error will either be corrected, or you will be explained why it is not an error.

§ 2. Molecular physics. Thermodynamics

Basic provisions of molecular kinetic theory(MCT) are as follows.
1. Substances consist of atoms and molecules.
2. Atoms and molecules are in continuous chaotic motion.
3. Atoms and molecules interact with each other with forces of attraction and repulsion
The nature of the movement and interaction of molecules can be different; in this regard, it is customary to distinguish between 3 states of aggregation of matter: solid, liquid and gaseous. The interactions between molecules are strongest in solids. In them, the molecules are located in the so-called nodes of the crystal lattice, i.e. in positions at which the forces of attraction and repulsion between molecules are equal. The motion of molecules in solids is reduced to vibrational motion around these equilibrium positions. In liquids, the situation is different in that, having oscillated around some equilibrium positions, the molecules often change them. In gases, molecules are far from each other, so the interaction forces between them are very small and the molecules move forward, occasionally colliding with each other and with the walls of the vessel in which they are located.
Relative molecular weight M r called the ratio of the mass m o of a molecule to 1/12 of the mass of a carbon atom m oc:

In molecular physics, the amount of a substance is usually measured in moles.
Molem ν is the amount of a substance that contains the same number of atoms or molecules (structural units) as there are in 12 g of carbon. This number of atoms in 12 g of carbon is called Avogadro's number:

Molar mass M = M r 10 −3 kg/mol is the mass of one mole of a substance. The number of moles in a substance can be calculated using the formula

The basic equation of the molecular kinetic theory of an ideal gas:

Where m 0- mass of the molecule; n- concentration of molecules; Ṽ - root mean square speed of molecules.

2.1. Gas laws

The equation of state of an ideal gas is the Mendeleev-Clapeyron equation:

Isothermal process(Boyle-Mariotte law):
For a given mass of gas at a constant temperature, the product of pressure and its volume is a constant:

In coordinates p−V isotherm is a hyperbola, and in coordinates V−T And p−T- straight (see Fig. 4)

Isochoric process(Charles' law):
For a given mass of gas at a constant volume, the ratio of pressure to temperature in degrees Kelvin is a constant value (see Fig. 5).

Isobaric process(Gay-Lussac's law):
For a given mass of gas at constant pressure, the ratio of gas volume to temperature in degrees Kelvin is a constant value (see Fig. 6).

Dalton's law:
If there is a mixture of several gases in a vessel, then the pressure of the mixture is equal to the sum of the partial pressures, i.e. those pressures that each gas would create in the absence of the others.

2.2. Elements of thermodynamics

Internal body energy equal to the sum of the kinetic energies of the random motion of all molecules relative to the center of mass of the body and the potential energies of interaction of all molecules with each other.
Internal energy of an ideal gas represents the sum of the kinetic energies of the random movement of its molecules; Since the molecules of an ideal gas do not interact with each other, their potential energy vanishes.
For an ideal monatomic gas, the internal energy is

Quantity of heat Q is a quantitative measure of the change in internal energy during heat exchange without performing work.
Specific heat- this is the amount of heat that 1 kg of a substance receives or gives up when its temperature changes by 1 K

Work in thermodynamics:
work during isobaric expansion of a gas is equal to the product of the gas pressure and the change in its volume:

Law of conservation of energy in thermal processes (first law of thermodynamics):
the change in the internal energy of a system during its transition from one state to another is equal to the sum of the work of external forces and the amount of heat transferred to the system:

Application of the first law of thermodynamics to isoprocesses:
A) isothermal process T = const ⇒ ∆T = 0.
In this case, the change in internal energy of an ideal gas

Hence: Q = A.
All the heat transferred to the gas is spent on doing work against external forces;

b) isochoric process V = const ⇒ ∆V = 0.
In this case, the gas work

Hence, ∆U = Q.
All heat transferred to the gas is spent on increasing its internal energy;

V) isobaric process p = const ⇒ ∆p = 0.
In this case:

Adiabatic is a process that occurs without heat exchange with the environment:

In this case A = −∆U, i.e. The change in the internal energy of the gas occurs due to the work done by the gas on external bodies.
When a gas expands, it does positive work. The work A performed by external bodies on a gas differs from the work done by a gas only in sign:

The amount of heat required to warm the body in a solid or liquid state within one state of aggregation, calculated by the formula

where c is the specific heat capacity of the body, m is the mass of the body, t 1 is the initial temperature, t 2 is the final temperature.
The amount of heat required to melt a body at the melting point, calculated by the formula

where λ is the specific heat of fusion, m is the mass of the body.
Amount of heat required for evaporation, calculated by the formula

where r is the specific heat of vaporization, m is the body mass.

In order to convert part of this energy into mechanical energy, heat engines are most often used. Heat engine efficiency is the ratio of the work A performed by the engine to the amount of heat received from the heater:

The French engineer S. Carnot came up with an ideal heat engine with an ideal gas as a working fluid. The efficiency of such a machine

Air, which is a mixture of gases, contains water vapor along with other gases. Their content is usually characterized by the term “humidity”. A distinction is made between absolute and relative humidity.
Absolute humidity is called the density of water vapor in the air - ρ ([ρ] = g/m3). Absolute humidity can be characterized by the partial pressure of water vapor - p([p] = mmHg; Pa).
Relative humidity (ϕ)- the ratio of the density of water vapor present in the air to the density of the water vapor that would have to be contained in the air at this temperature for the vapor to be saturated. Relative humidity can be measured as the ratio of the partial pressure of water vapor (p) to the partial pressure (p0) that saturated vapor has at that temperature:

Goal: repetition of basic concepts, laws and formulas of molecular physics in accordance with the Unified State Examination codifier

Content elements tested at the Unified State Exam 2012:
1.Basic provisions of the ICT.
2. Models of the structure of gases, liquids and solids.
3. Ideal gas model.
4. Basic equation of MKT of an ideal gas.
5. Absolute temperature as a measure of its average kinetic energy
particles.
6. Mendeleev-Clapeyron equation.
7. Isoprocesses.
8. Mutual transformations of liquids and gases.
9.Saturated and unsaturated pairs. Air humidity.
10. Changes in the aggregate states of matter. Melting and
hardening.
11.Thermodynamics: internal energy, amount of heat, work.
12.First law of thermodynamics
13.Second law of thermodynamics.
14. Application of the first law of thermodynamics to isoprocesses.
15.Efficiency of heat engines.

Basic provisions of the ICT

The molecular kinetic theory is called
the study of the structure and properties of matter based on
ideas about the existence of atoms and molecules as
the smallest particles of a chemical substance.
Main provisions of the ICT:
1. All substances - liquid, solid and gaseous -
formed from tiny particles - molecules,
which themselves are made of atoms.
2. Atoms and molecules are in continuous
chaotic movement.
3. Particles interact with each other by forces,
having an electrical nature (they are attracted and
repulse).

Atom. Molecule.

An atom is the smallest
part of the chemical
element having
its properties,
capable of
independent
existence.
Molecule –
smallest stable
particle of matter
made up of atoms
one or more
chemical elements,
preserving the basic
Chemical properties
of this substance.

Mass of molecules. Amount of substance.

Relative molecular (or atomic)
the mass of a substance is called the ratio
masses
m0
M r of substance to 1/12
molecule (or atom) of a given
1
mass of carbon atom 12C.
m0C
The amount of substance is 12
number of molecules in
body, but expressed in relative units.
A mole is the amount of substance containing
as many particles (molecules) as atoms
contained in 0.012 kg of carbon 12C.
23
1
Means
any
substances contained
N A 6c 110mol
mole
the same number of particles (molecules). This number
is called Avogadro's constant NА.
The amount of substance is equal to N the ratio of the number
molecules in a given body to a constant
Avogadro, i.e.
N.A.
to the number of molecules in 1 mole of a substance.
kg
3
m
MM
M
r 10
m0 N A
The molar mass of a substance is called
mass
mole
substance taken in an amount of 1 mol.

Molecules of most solids
arranged in a certain order.
Such solids are called
crystalline.
Particle movements are
oscillations around equilibrium positions.
If we connect the centers of positions
equilibrium of particles, then it turns out
correct spatial lattice,
called crystalline.
The distances between molecules are comparable
with molecular sizes.
Main properties: retain shape and
volume. Single crystals are anisotropic.
Anisotropy – dependence of physical
properties depending on the direction in the crystal.
l r0

Models of the structure of solids, liquids and gases

Distances between molecules
liquids comparable in size
molecules, so there is little liquid
shrinks.
Liquid molecule vibrates
near the position of the temporary
balance when facing others
molecules from the nearest
environment. From time to time she
manages to make the jump
to keep doing
fluctuations among other neighbors.
"Jumping" of molecules occurs along
in all directions with the same
frequency, this explains
fluidity of a liquid and what it
takes the shape of a vessel
l r0

Models of the structure of solids, liquids and gases

Distance between gas molecules
much larger than themselves
molecules, so gas can be compressed so
that its volume will decrease by several
once.
Molecules with enormous speeds
moving in the space between
collisions. During
collisions change molecules dramatically
speed and direction of movement.
Molecules attract very weakly
to each other, so the gases do not have
own form and constant
volume.
l r0

Thermal motion of molecules

Erratic chaotic movement
molecules is called thermal
movement. Proof
thermal movement is
Brownian motion and diffusion.
Brownian motion is thermal
movement of tiny particles
suspended in liquid or gas,
occurring under the influence of blows
molecules of the environment.
Diffusion is a phenomenon
penetration of two or more
substances in contact with each other
friend.
The rate of diffusion depends on
aggregative state of matter and
body temperature.

10. Interaction of particles of matter

Interaction forces between molecules.
At very small distances between molecules
Repulsive forces are necessarily at work.
At distances exceeding 2 - 3 diameters
molecules, attractive forces act.

11. Ideal gas model

Ideal gas is a theoretical model
gas, in which the dimensions and
interactions of gas particles and take into account
only their elastic collisions.
In the kinetic model of an ideal gas
molecules are considered ideal
elastic balls interacting between
with oneself and with the walls only during elastic
collisions.
The total volume of all molecules is assumed
small compared to the volume of the vessel, in
where the gas is located.
Colliding with the wall of a container, gas molecules
put pressure on her.
Microscopic parameters: mass,
speed, kinetic energy of molecules.
Macroscopic parameters: pressure,
volume, temperature.

12. Basic equation of MCT gases

The pressure of an ideal gas is two thirds
average kinetic energy of translational
movement of molecules contained in a unit volume
where n = N / V – concentration of molecules (i.e. number
molecules per unit volume of the vessel)
Dalton's law: pressure in a mixture is chemically
of non-interacting gases is equal to their sum
partial pressures
p = p1 + p2 + p3

13. Absolute temperature

Temperature characterizes the degree of heating of the body.
Thermal equilibrium is a state of the system
bodies in thermal contact, in which there is no
heat transfer occurs from one body to another, and
all macroscopic parameters of bodies remain
unchanged.
Temperature is a physical parameter that is the same
for all bodies in thermal equilibrium.
To measure temperature, physical
devices - thermometers.
There is a minimum possible temperature at
which stops the chaotic movement of molecules.
It is called absolute zero temperature.
The Kelvin temperature scale is called absolute
temperature scale.
T t 273

14. Absolute temperature

Average kinetic energy of chaotic motion
gas molecules is directly proportional to the absolute
temperature.
3
E kT
2
2
p nE p nkT
3
k – Boltzmann constant – relates the temperature in
energy units with temperature in kelvins
Temperature is a measure of average kinetic energy
translational movement of molecules.
At the same pressures and temperatures, the concentration
molecules are the same for all gases
Avogadro's law: in equal volumes of gases at equal
temperatures and pressures contain the same number
molecules

15. Mendeleev-Clapeyron equation

The ideal gas equation of state is the relationship between
ideal gas parameters - pressure, volume and
absolute temperature that determines its state.
pV RT
m
RT
M
R kN A 8.31
J
mole K
R is the universal gas constant.
Avogadro's law: one mole of any gas under normal conditions
occupies the same volume V0, equal to 0.0224 m3/mol.
From the equation of state follows the relationship between pressure,
volume and temperature of an ideal gas that can
be in any two states.
Clapeyron's equation
pV
pV
1 1
T1
2 2
T2
const.

16. Isoprocesses

Isoprocesses are processes in which
one of the parameters (p, V or T) remains
unchanged.
Isothermal process (T = const) –
state change process
thermodynamic system flowing
at constant temperature T.
Boyle–Mariotte law: for a given gas
mass product of gas pressure and its
volume is constant if the gas temperature is not
is changing.
const
pV const p
V
T3 > T2 > T1

17. Isoprocesses

An isochoric process is a process of change

constant volume.
Charles's law: for a gas of a given mass
the ratio of pressure to temperature is constant,
if the volume does not change.
p
const p const T
T
V3 > V2 > V1

18. Isoprocesses

An isobaric process is a process of change
state of the thermodynamic system at
constant pressure.
Gay-Lussac's law: for a gas of a given mass
The ratio of volume to temperature is constant if
the gas pressure does not change.
V
V V0 1 t
const V const T
T
At constant pressure, the volume of an ideal gas is
varies linearly with temperature.
where V0 is the volume of gas at a temperature of 0 °C.
α = 1/273.15 K–1 - volumetric temperature coefficient
expansion of gases.
p3 > p2 > p1

19. Mutual transformations of liquids and gases

Vaporization is the transition of a substance from
liquid state into gaseous state.
Condensation is the transition of a substance from
gaseous state into liquid.
Evaporation is the formation of vapor
originating from a free surface
liquids.
From a molecular kinetic point of view
theory, evaporation is a process in which
liquid surfaces fly off the most
fast molecules, kinetic energy
which exceeds the energy of their connection with
the remaining molecules of the liquid. This leads
to a decrease in average kinetic energy
remaining molecules, i.e. to cooling
liquids.
During condensation, there is a release
some amount of heat into the environment
Wednesday.

20. Mutual transformations of liquids and gases Saturated and unsaturated vapors

In a closed container there is liquid and its
steam may be in a state
dynamic equilibrium when
number of molecules leaving
liquid is equal to the number of molecules
returning to the liquid from
steam, i.e. when the speed of processes
evaporation and condensation
are the same.
Steam in equilibrium with
its liquid is called
saturated.
Saturated vapor pressure p0
of this substance depends only on
its temperature and does not depend on
volume
Saturated vapor pressure increases
not only as a result of the increase
liquid temperature, but also
due to increase
concentration of vapor molecules.
p0 nkT

21. Mutual transformations of liquids and gases Boiling

Boiling is vaporization
occurring throughout the entire volume of liquid.
The liquid begins to boil at
such a temperature at which
its saturated vapor pressure
becomes equal to the pressure in
liquid, which is made up of
air pressure on the surface
fluids (external pressure) and
column hydrostatic pressure
liquids.
Each liquid has its own temperature
boiling point, which depends on pressure
saturated steam. The lower the pressure
saturated steam, the higher
boiling temperature corresponding
liquids

22. Humidity

Humidity is the content of water in the air
pair.
The more water vapor is in a certain volume
air, the closer the steam is to the saturation state. The higher
air temperature, the greater the amount of water vapor
required for its saturation.
Absolute humidity is the density of water vapor
expressed in kg/m3 or its partial pressure - pressure
water vapor it would produce if all the other
there were no gases.
Relative air humidity is the ratio
absolute air humidity to saturated vapor density
at the same temperature or is it the ratio of partial
vapor pressure in air to saturated vapor pressure at that
same temperature.
p
100%;
100%
0
p0
Hygrometers are used to determine air humidity:
condensation and hair; and a psychrometer.

23. Change in aggregative states of matter: melting and crystallization

Melting is the transition of a substance from
solid to liquid state.
Solidification or crystallization - the transition of a substance from a liquid state to
solid.
The temperature at which a substance
it starts to melt, it's called
melting temperature.
During the melting of its substance
the temperature does not change, because energy,
received by the substance is spent on
destruction of the crystal lattice. At
solidification forms a crystalline
lattice, in this case energy is released and
the temperature of the substance does not change.
Amorphous bodies do not have a specific
melting temperature.

24. Thermodynamics

Thermodynamics is the theory of thermal processes,
which does not take into account the molecular structure
tel.
Basic concepts of thermodynamics:
Macroscopic system is a system consisting
from a large number of particles.
Closed system - a system isolated from
any external influences.
The equilibrium state is the state
macroscopic system, in which
parameters characterizing its condition,
remain unchanged in all parts of the system.
A process in thermodynamics is called
change in body condition over time.

25. Internal energy

The internal energy of a body is the sum
kinetic energy of all its molecules and
potential energy of their interaction.
Internal energy of an ideal gas
determined only by kinetic energy
its random forward movement
molecules.
3 m
3
U
RT
U pV
2M
2
Internal energy of an ideal monatomic
of a gas is directly proportional to its temperature.
Internal energy can be changed by two
ways: doing work and
heat transfer.

26. Heat transfer

Heat transfer is
spontaneous transmission process
heat occurring between bodies
with different temperatures.
Types of Heat Transfer
Thermal conductivity
Convection
Radiation

27. Amount of heat

The amount of heat is called
quantitative measure of change
internal energy of the body at
heat exchange (heat transfer).

heating the body or emitted by it
when cooling:
с – specific heat capacity –
physical quantity showing
how much heat is required
for heating 1 kg of substance by 1 0C.
The amount of heat released when
complete combustion of fuel.
q – specific heat of combustion –

amount of heat released when
complete combustion of fuel weighing 1 kg.
Q cm t2 t1
Qqm

28. Amount of heat

The amount of heat required for
melting of a crystalline body or
secreted by the body during hardening.
λ – specific heat of fusion –
a value indicating what
amount of heat needed
inform the crystalline body
weighing 1 kg, so that at a temperature
melting completely convert it into
liquid state.
The amount of heat required for
complete transformation of liquid
substances vaporized or released by the body
during condensation.
r or L – specific heat
vaporization – value,
showing how much
heat is needed to convert
liquid weighing 1 kg in steam without
temperature changes.
Qm
Q rm; Q Lm

29. Work in thermodynamics

In thermodynamics, unlike mechanics,
it is not the movement of the body as a whole that is considered,
but only moving parts
macroscopic body relative to each other
friend. As a result, the volume of the body changes, and
its speed remains zero.
When expanding, the gas makes
positive work A" = pΔV. Work A,
performed by external bodies above a gas
differs from the work of gas A" only by the sign: A
= - A".
On the graph of pressure versus volume
work is defined as the area of a figure under
schedule.

30. First law of thermodynamics

The first law of thermodynamics is the law of conservation and
energy conversion for a thermodynamic system.
Change in the internal energy of the system during its transition
from one state to another is equal to the amount of work
external forces and the amount of heat transferred to the system.
U A Q
If the work is done by the system and not by external forces:
Q U A
The amount of heat transferred to the system goes to
change in its internal energy and to perform
system of working on external bodies.

31. Application of the first law of thermodynamics to various processes

Isobaric process.
The amount of heat transferred to the system is
Q U A
goes to change its internal energy and
the system performs work on external
bodies.
Isochoric process: V – const => A = 0
The change in internal energy is
the amount of heat transferred.
Isothermal process: T – const => ΔU = 0
The entire amount of heat transferred to the gas goes
to complete the work.
Adiabatic process: occurs in a system
which does not exchange heat with
surrounding bodies, i.e. Q = 0
The change in internal energy occurs
only by doing work.
U Q
Q A
U A

32. Second law of thermodynamics

All processes occur spontaneously in
one specific direction. They
irreversible. Warmth always comes from
hot body to cold, and mechanical
the energy of macroscopic bodies - into the internal one.
The direction of processes in nature indicates
second law of thermodynamics.
R. Clausius (1822 – 1888): impossible
transfer heat from a colder system to
hotter in the absence of others
simultaneous changes in both systems or
in surrounding bodies.

33. Efficiency of a heat engine

Heat engines – devices,
converting internal energy
fuel to mechanical.
The working fluid of all TDs is gas,
which is obtained during fuel combustion
quantity of heat Q1, makes
work A" during expansion. Part
heat Q2 is inevitably transferred
refrigerator, i.e. gets lost.
Efficiency factor
a heat engine is called
ratio of work done
engine, to the amount of heat,
received from the heater:
An ideal Carnot heat engine with
ideal gas as a working gas
body has the maximum possible
Efficiency:
A Q1 Q2
A Q1 Q2
Q1
Q1
max
T1 T2
T1

34.

35.

1. The thermometer is not designed for high temperatures
and requires replacement
2. The thermometer shows higher
temperature
3. The thermometer shows a lower temperature
4. The thermometer shows the calculated temperature

36.

1. 180C.
2. 190С
3. 210C.
4. 220C.

37.

T,K
350
300
0
t(min)
2
4
6
8
1. The heat capacity of water increases with time
2. after 5 minutes all the water has evaporated
3. at a temperature of 350 K, water gives off so much heat to the air,
how much does he get from gas?
4. after 5 minutes the water begins to boil

38.

1. Water moves from
solid state in
liquid at 00C.
2. Water boils at 1000C.
3. Heat capacity of water
equal to 4200 J/(kg 0C).
4. The longer it takes to heat up
water, the higher it is
temperature.

39.

1. In position I, heat transfer occurs from body 1 to body 2.
2. In position II, heat transfer occurs from body 1 to body 2.
3. In any position, heat transfer occurs from body 2
to the body 1.
4. Heat transfer occurs only in position II.

40.

R
R
P
R
50
50
50
50
(IN)
40
40
(A)
(B)
30
(G)
40
30
30
20
20
20
10
10
10
0
0
0
0
2
4
6
8
2
4
6
8
10
00
10
2
4
6
8
10
10
1) Schedule A
V
V
V
2) Schedule B
3) Schedule B
V
4) Schedule G.

41.

1. only A
2. only B
3. only B
4. A, B and C

42.

E k
1
1. 1
2. 2
3. 3
4. 4
1
2
3
4
0
T

43.

44.

1. A
2. B
3. B
4. G
P, kPa
A
B
2
IN
1
0
G
1
2
3
V,m

45.

1. equal to the average kinetic energy of molecules
liquids
2. exceeds average kinetic energy
liquid molecules
3. less than the average kinetic energy of molecules
liquids
4. equal to the total kinetic energy of molecules
liquids

46.

1. Increased 4 times
2. Decreased by 2 times
3. Increased by 2 times
4. Has not changed
pV
const T
const p
T
V

47.

48.

1.
2.
3.
4.
200 K
400 K
600 K
1200 K
P, kPa
200
100
0
2
1
4
1
3
2
3
3 V, m
p4V4 p2V2
p2V2
200 3 200
T2
T4
1200 K
T4
T2
p4V4
100 1

49.

1.
2.
3.
4.
decreased by 3 times
increased 3 times
increased 9 times
hasn't changed
2
pnE
3

50.

1.
2.
3.
4.
isobaric heating
isochoric cooling
isothermal compression
isochoric heating

51.

1. heater power
2. substances of the vessel in which water is heated
3. atmospheric pressure
4. initial water temperature

3. when high, since this causes sweat

64.

1.
2.
3.
4.
only in liquid state
only in solid state
in both liquid and solid states
in both liquid and gaseous states

65.

FEATURES OF THE ISOPROCESS
NAME
ISOPROCESS
A) The entire amount of heat transferred to the gas goes to
work done, and the internal energy of the gas
remains unchanged.
1) isothermal
B) The internal energy of the gas changes
only by doing work, since
there is no heat exchange with surrounding bodies.
2) isobaric
3) isochoric
4) adiabatic
A
B
1
4

66.

1
2
3

67.

1. After placing the jar on the fire, the water in it
heated through the thin wall of the jar from hot
gas combustion products. Moreover, with increasing temperature
the water evaporated and its vapor pressure increased
jar, which gradually displaced the air from it.
When the water boiled and almost all evaporated, the air
There is practically nothing left inside the jar. Pressure
saturated vapor in the jar became equal to
external atmospheric pressure.
2. When the jar was removed from the heat, closed with a lid and cooled
cold water to almost room temperature,
hot water vapor inside the jar has cooled down and is almost
completely condensed on its walls, giving
heat of condensation to the outside, cold water, thanks to
the process of heat conduction through the walls.

68.

1. In accordance with the Clapeyron–Mendeleev equation
2.
the steam pressure in the jar dropped sharply - firstly, due to
reducing the mass of steam remaining in the can, and, secondly,
due to a drop in its temperature. Note that sharp
the decrease in pressure in the bank can be explained this way: when
When the temperature drops to room temperature, they condense,
remaining saturated, but their pressure becomes much
less than the saturated vapor pressure of water at temperature
boiling (about 40 times).
Since at room temperature the saturated pressure
water vapor is only a small fraction of atmospheric
pressure (no more than 3–4%), a thin jar after watering it
water will be under the influence of the difference of this large
external pressure and low vapor pressure inside. By this
cause large compressive forces will begin to act on the jar
forces that will tend to flatten the jar. As soon as
these forces will exceed the maximum value that can
withstand the walls of the jar, it will flatten and sharply
will decrease in volume.

69.

According to the first law
thermodynamics the amount of heat,
required to melt ice, ΔQ1
= λm, where λ is specific heat
melting ice. ΔQ2 – supplied
Joule heat: ΔQ2 = ηPt. IN
according to specified conditions
ΔQ1 = 66 kJ and ΔQ2 = 84 kJ, which means
ΔQ1< ΔQ2, и поставленная задача
doable

70.

According to the first law of thermodynamics, the quantity
heat Q transferred to the gas goes to change it
internal energy ΔU and work done by this gas
A, that is, Q = ΔU + A. When the gas is heated,
its isobaric expansion. In this process, gas work
is equal to A = pΔV, where the change in gas volume ΔV = Sl = πR2l.
From the piston equilibrium condition (see figure) we find
gas pressure: pS = p0S + Mgcosα, from where
Mgcos
p p0
S
Then the required value is
Mgcos
U Q R l p0
2
R
2

71.

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real tasks of the Unified State Exam 2010, Physics [Text]: a textbook for
graduates. Wed textbook establishments / A.V. Berkov, V.A. Gribov. - OOO
"Astrel Publishing House", 2009. – 160 p.
2. Kasyanov, V.A. Physics, grade 11 [Text]: textbook for
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116 p.
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http://fipi.ru/view/sections/92/docs/
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