The kinetic theory of gases

Сентябрь 2, 2022

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The kinetic theory of gases

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2. Course of lectures «Contemporary Physics: Part1» Lecture №8 The Kinetic Theory of Gases. Heat Engines, Entropy,
3. Molecular Model of an Ideal Gas The macroscopic description of model of ideal gas make the
4. Molecular Model of an Ideal Gas
5. Molecular Model of an Ideal Gas
6. Molecular Model of an Ideal Gas (10.1)
7. Molecular Model of an Ideal Gas
8. Molecular Model of an Ideal Gas (10.2) The pressure of a gas is proportional to the
9. Molecular Interpretation of Temperature This result tells us that temperature is a direct measure of average
10. Molecular Interpretation of Temperature Theorem of equipartition of energy each degree of freedom contributes ½kbT to
11. Molecular Interpretation of Temperature Table 10.1
12. Molar Specific Heat of an Ideal Gas molar specific heats: (10.8) (10.9) (10.10)
13. Molar Specific Heat of an Ideal Gas If energy is transferred by heat to a system
14. Molar Specific Heat of an Ideal Gas (10.15)
15. Molar Specific Heat of an Ideal Gas (10.16) (10.17)
16. Adiabatic Processes for an Ideal Gas An adiabatic process is one in which no energy is
17. The Boltzmann Distribution Law (10.7) The number density (10.19) The Boltzmann distribution law, is important in
18. Distribution of Molecular Speeds (10.20)
19. Distribution of Molecular Speeds
20. Mean Free Path The average distance between collisions is called the mean free path.
21. Mean Free Path
22. Mean Free Path (10.21) (10.22) Collision frequency f, is
23. Heat Engines and the Second Law of Thermodynamics A heat engine is a device that takes
24. Heat Engines and the Second Law of Thermodynamics The net work done in a cyclic process
25. On the basis of the fact that efficiencies of real engines are well below 100%, the
26. Heat Pumps and Refrigerators
27. Heat Pumps and Refrigerators The Clausius statement states: It is impossible to construct a cyclical machine
28. Reversible and Irreversible Processes In a reversible process, the system undergoing the process can be returned
29. The Carnot Engine French engineer named Sadi Carnot showed that a heat engine operating in an
31. The Carnot Engine This result indicates that all Carnot engines operating between the same two temperatures
32. Entropy Isolated systems tend toward disorder and that entropy is a measure of this disorder. We
33. Entropy The change in entropy during a process depends only on the end points and therefore
34. Entropy Let us consider the changes in entropy that occur in a Carnot heat engine that
35. Quick Quiz 5.3 If you are asked to make a very sensitive glass thermometer, which of
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Course of lectures «Contemporary Physics: Part1»
Lecture №8
The Kinetic Theory of Gases.
Heat

Engines, Entropy, and the Second Law of Thermodynamics.

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Molecular Model of an Ideal Gas
The macroscopic description of model of

ideal gas make the following assumptions:
1. The number of molecules in the gas is large, and the average separation between them is large compared with their dimensions.
2. The molecules obey Newton’s laws of motion, but as a whole they move randomly.
3. The molecules interact only by short-range forces during elastic collisions.
4. The molecules make elastic collisions with the walls.
5. The gas under consideration is a pure substance; that is, all molecules are identical.

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Molecular Model of an Ideal Gas

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Molecular Model of an Ideal Gas

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Molecular Model of an Ideal Gas
(10.1)

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Molecular Model of an Ideal Gas

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Molecular Model of an Ideal Gas
(10.2)
The pressure of a gas is

proportional to the number of molecules per unit volume and to the average translational kinetic energy of the molecules,

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Molecular Interpretation of Temperature
This result tells us that temperature is a

direct measure of average molecular kinetic energy.

(10.3)

(10.4)

(10.5)

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Molecular Interpretation of Temperature
Theorem of equipartition of energy
each degree of freedom

contributes ½kbT to the energy of a system, where possible degrees of freedom in addition to those associated with translation arise from rotation and vibration of molecules.

(10.7)

(10.6)

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Molecular Interpretation of Temperature
Table 10.1

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Molar Specific Heat of an Ideal Gas
molar specific heats:
(10.8)
(10.9)
(10.10)

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Molar Specific Heat of an Ideal Gas
If energy is transferred by

heat to a system at constant volume, then no work is done on the system. From the first law of thermodynamics, we see that

(10.11)

(10.12)

(10.13)

(10.14)

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Molar Specific Heat of an Ideal Gas
(10.15)

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Molar Specific Heat of an Ideal Gas
(10.16)
(10.17)

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Adiabatic Processes for an Ideal Gas
An adiabatic process is one in

which no energy is transferred by heat between a system and its surroundings.

(10.18)

Home work:

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The Boltzmann Distribution Law
(10.7)
The number density
(10.19)
The Boltzmann distribution law, is important

in describing the statistical mechanics of a large number of molecules. It states that the probability of finding the molecules in a particular energy state varies exponentially as the negative of the energy divided by kBT.

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Distribution of Molecular Speeds
(10.20)

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Distribution of Molecular Speeds

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Mean Free Path
The average distance between collisions is called the mean

free path.

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Mean Free Path

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Mean Free Path
(10.21)
(10.22)
Collision frequency f, is

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Heat Engines and the Second Law
of Thermodynamics
A heat engine is a

device that takes in energy by heat and, operating in a cyclic process, expels a fraction of that energy by means of work.

The net work Weng done by a heat engine is equal to the net energy Qnet transferred to it.

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Heat Engines and the Second Law
of Thermodynamics
The net work done in

a cyclic process is the area enclosed by the curve representing the process on a PV diagram.

(10.23)

The thermal efficiency

(10.24)

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On the basis of the fact that efficiencies of real engines

are well below 100%, the Kelvin–Planck form of the second law of thermodynamics states the following:

Heat Engines and the Second Law
of Thermodynamics

It is impossible to construct a heat engine that, operating in a cycle, produces no effect other than the input of energy by heat from a reservoir and the performance of an equal amount of work.

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Heat Pumps and Refrigerators

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Heat Pumps and Refrigerators
The Clausius statement states:
It is impossible to construct

a cyclical machine whose sole effect is to transfer energy continuously by heat from one object to another object at a higher temperature without the input of energy by work.

In simpler terms, energy does not transfer spontaneously by heat from a cold object to a hot object.

The effectiveness of a heat pump is described in terms of a number called the coefficient of performance (COP).

(10.25)

(10.26)

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Reversible and Irreversible Processes
In a reversible process, the system undergoing the

process can be returned to its initial conditions along the same path on a PV diagram, and every point along this path is an equilibrium state. A process that does not satisfy these requirements is irreversible.

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The Carnot Engine
French engineer named Sadi Carnot showed that a heat

engine operating in an ideal, reversible cycle— called a Carnot cycle—between two energy reservoirs is the most efficient engine possible.

Carnot’s theorem can be stated as follows:

No real heat engine operating between two energy reservoirs can be more efficient than a Carnot engine operating between the same two reservoirs.

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The Carnot Engine
This result indicates that all Carnot engines operating between

the same two temperatures have the same efficiency.

Hence, the thermal efficiency of a Carnot engine is

(10.27)

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Entropy
Isolated systems tend toward disorder and that entropy is a measure

of this disorder.

We distinguish between microstates and macrostates of a system. A microstate is a particular configuration of the individual constituents of the system.
Macrostate is a description of the conditions of the system from a macroscopic point of view and makes use of macroscopic variables such as pressure, density, and temperature for gases.

Because entropy is a measure of disorder, an alternative way of stating this is the entropy of the Universe increases in all real processes.

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Entropy
The change in entropy during a process depends only on the

end points and therefore is independent of the actual path followed. Consequently, the entropy change for an irreversible process can be determined by calculating the entropy change for a reversible process that connects the same initial and final states.

(10.28)

(10.29)

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Entropy
Let us consider the changes in entropy that occur in a

Carnot heat engine that operates between the temperatures Tc and Th. In one cycle, the engine takes in energy Qh from the hot reservoir and expels energy Qc to the cold reservoir.

(10.30)

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