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Physics Lesson 13.8.2 - Molar Specific Heat at Constant Volume

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Welcome to our Physics lesson on Molar Specific Heat at Constant Volume, this is the second lesson of our suite of physics lessons covering the topic of Molar Specific Heats and Degrees of Freedom, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

Molar Specific Heat at Constant Volume

By definition, Molar specific heat of an ideal gas is the heat we provide to the gas to raise the temperature of one mole through 1K or 1°C. It differs from the specific heat capacity c discussed earlier, as the molar specific heat is calculated for 1 mole instead of 1 kg of material. We represent it as C and its unit is J mol/K.

Let's consider a gas inside a fixed cylinder, as the one shown in the figure below.

Physics Tutorials: This image provides visual information for the physics tutorial Molar Specific Heats and Degrees of Freedom

If the amount of gas inside the container is n moles, its initial temperature is T, pressure P and volume V, we consider these parameters as those of the initial state i.

If we provide a very small amount of heat energy to the gas, its temperature increases by ΔT and pressure increases by ΔP as well. Thus, the gas is brought to a final state f, which has new values for the gas parameters.

Given that volume is constant, we obtain for the P - V graph of the two states:

Physics Tutorials: This image provides visual information for the physics tutorial Molar Specific Heats and Degrees of Freedom

The heat absorbed by the gas during this process therefore is

Q = n × Cv × ∆T

where CV is the molar specific heat at constant volume.

From the First Law of Thermodynamics, we know that

Q = ∆U + W

Rearranging, we obtain

∆U = Q - W

Thus, substituting the above value of Q, we obtain

∆U = n × Cv × ∆T - W

Since the volume of gas is constant, there is no work done on or by the gas. Hence,

∆U = n × Cv × ∆T

Or

Cv = ∆U/n × ∆T

Substituting

∆U = 3/2 × n × R × ∆T

in the above equation, we obtain

Cv = 3/2 × n × R × ∆T/n × ∆T

Simplifying, we obtain for the molar specific heat of ideal gas

Cv = 3/2 R

Since R = 8.31 J/K, we obtain for the value of molar specific heat of a monoatomic ideal gas:

Cv = 3/2 × 8.31 J/mol × K
≈ 12.5 J/mol × K

When dealing with two or more atomic gases, we observe that the motion of molecules is not that simple, as atoms in monoatomic gases, because molecules can also spin or vibrate, not only move in a translational way. Therefore, the value of CV will be greater than that obtained above for monoatomic gases.

Summarizing everything discussed above, we conclude that:

The actual internal energy of an ideal gas at constant volume is

U = n × Cv × T

and the change in the internal energy of an ideal gas at constant volume is

∆U = n × Cv × ∆T

Example 1

0.3 moles of a monoatomic ideal gas are at 27°C inside a fixed container.

  1. What is the value of actual internal energy of this gas?
  2. By how many joules will the internal energy of gas increase when the temperature becomes 177°C?
  3. What will be the value of final internal energy of gas?

Solution 1

Clues:

n = 0.3 moles
T1 = 27°C = 27 + 273 = 300 K
T2 = 177°C = 177 + 273 = 450 K
Cv = 12.5 J/molK

Hence, we have

a) The actual (initial) internal energy of gas is

Ui = n × Cv × T
= 0.3 mol × 12.5 J/mol × K × 300 K
= 1125 J

b) Then increase in the internal energy of gas is

∆U = n × Cv × ∆T
= 0.3 mol × 12.5 J/mol × K × (450 - 300)K
= 562.5 J

c) The value of final internal energy of gas will be

Uf = Ui + ∆U
= 1125 J + 562.5 J
= 1687.5 J

The graph below shows three different paths representing three different processes that bring the ideal gas from temperature T to another temperature T + ΔT (one is at constant volume, one at constant pressure and the other is at changing volume and pressure).

Physics Tutorials: This image provides visual information for the physics tutorial Molar Specific Heats and Degrees of Freedom

It is obvious that the change in temperature is the same in all paths shown above and in many others. This means the change in internal energy is independent from the path followed during the process. Thus, we reached in a very important conclusion:

Any change in the internal energy of an ideal gas enclosed within a container depends only on the change in temperature; it is independent from the type of process that has brought that change.

You have reached the end of Physics lesson 13.8.2 Molar Specific Heat at Constant Volume. There are 4 lessons in this physics tutorial covering Molar Specific Heats and Degrees of Freedom, you can access all the lessons from this tutorial below.

More Molar Specific Heats and Degrees of Freedom Lessons and Learning Resources

Thermodynamics Learning Material
Tutorial IDPhysics Tutorial TitleTutorialVideo
Tutorial		Revision
Notes		Revision
Questions
13.8Molar Specific Heats and Degrees of Freedom
Lesson IDPhysics Lesson TitleLessonVideo
Lesson
13.8.1Internal Energy
13.8.2Molar Specific Heat at Constant Volume
13.8.3Molar Specific Heat at Constant Pressure
13.8.4Degrees of Freedom and Molar Specific Heats

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