Question

Relating \(C_{V}\) and \(C_{p} .\) Show that \(C_{p}=C_{V}+N k\) for an ideal gas.

Short answer

\[C_P = C_V + Nk\]

Step-by-step solution

- Understand the Relation Between Heat Capacities

For an ideal gas, the heat capacities at constant volume (\(C_V\)) and constant pressure (\(C_P\)) are related through the equation: \[C_P = C_V + nR\] where \(n\) is the number of moles and \(R\) is the universal gas constant.

- Express in Terms of Boltzmann Constant

We know that the universal gas constant \(R\) can be related to the Boltzmann constant \(k\) by the equation: \[R = kN_A\] where \(N_A\) is Avogadro's number.

- Substitute \(R\) in the Heat Capacity Equation

By substituting \(R = kN_A\) into the heat capacity equation, we get: \[C_P = C_V + n k N_A\]

- Convert to Number of Particles \(N\)

Remember that the number of moles \(n\) times Avogadro's number \(N_A\) equals the total number of particles \(N\): \[N = nN_A\] so the equation becomes: \[C_P = C_V + Nk\]

- State the Final Relation

Therefore, for an ideal gas, the relation between heat capacities at constant pressure and volume is given by: \[C_P = C_V + Nk\]

Key concepts

Heat Capacity at Constant Volume

Heat capacity at constant volume (\(C_V\)) is a key concept in thermodynamics. It refers to the amount of heat energy required to raise the temperature of a given quantity of gas by one degree while the volume is kept constant. For an ideal gas, this concept is quite straightforward because the internal energy change is only due to temperature change.
Whenever we heat a gas at constant volume, no work is done because the volume doesn't change. Thus, all the heat added goes into increasing the internal energy of the gas. The formula for heat capacity at constant volume is:
\[ C_V = \frac{dU}{dT} \] where \( dU \) is the change in internal energy and \( dT \) is the change in temperature. For an ideal monatomic gas, \( C_V = \frac{3}{2}nR \) where \( n \) is the number of moles and \( R \) is the universal gas constant.

Heat Capacity at Constant Pressure

Heat capacity at constant pressure (\(C_P\)) differs from heat capacity at constant volume in that it takes into account the work done by the gas while it expands. When heating a gas at constant pressure, not only does the internal energy increase, but the gas also does work on its surroundings by expanding.
Thus, the formula for heat capacity at constant pressure is slightly different:
\[ C_P = \frac{dQ}{dT} \] where \( dQ \) is the heat added and \( dT \) is the change in temperature. For an ideal monatomic gas, \( C_P = \frac{5}{2}nR \) .
Notice how \( C_P \) is always greater than \( C_V \) due to the additional work done in expanding the gas.
Using the relationship \[ C_P = C_V + nR \] and knowing \( R \) is the gas constant, we can understand that extra energy, linked to the constant pressure process, can relate these two heat capacities.

Boltzmann Constant

The Boltzmann constant (\(k\)) is an essential fundamental constant in physics that relates the average kinetic energy of particles in a gas to the temperature of the gas. It bridges the macroscopic and microscopic worlds, connecting temperature with energy at the particle level.
The Boltzmann constant can be defined by the equation:
\[ k = 1.38 \times 10^{-23} \text{ J/K} \]
where \( \text{J} \) is Joules and \( \text{K} \) is Kelvin. This constant is crucial in many areas of physics, especially in statistical mechanics and thermodynamics.
For ideal gases, it also plays an important role in understanding the heat capacities. The universal gas constant \( R \) is actually the Boltzmann constant multiplied by Avogadro's number (\(N_A\)):
\[ R = k N_A \]
This means the relations between \( C_P \) and \( C_V \) can also be explored using \( k \). This leads us to the relationship: \[ C_P = C_V + Nk \] where \( N \) is the total number of particles in the gas.
This demonstrates the fundamental nature of the Boltzmann constant in linking macroscopic thermodynamic properties with the molecular scale.