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CP – CV for Non–Ideal Gases

What is a Non-Ideal Gas Behaviour?

Non-ideal gas behaviour is the observed relationships between the temperature, volume, and pressure of a gas that are not properly or accurately described by the ideal gas laws. When the intermolecular attractions between the molecules of the gases are negligible, the ideal gas equation works appropriately. Furthermore, the gas molecules have no volume compared to the total volume. When the pressure is low, and the temperature is high, this happens.

If the pressures and temperatures are not the same, the ideal gas law may not produce accurate results. As a result, we have a non-ideal situation, and we can also say that gases do not always behave ideally.

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Derivation of CP – CV for Non–Ideal Gases

To derive a relationship for CP – CV for a non-ideal gas, we need to know the following terms, which are as follows-

  • Maxwell’s Relations
  • Basic Thermodynamic Equations


dQ = TdS (eq. 1)

where Q is the heat given to the system, T is the temperature and S is the entropy of the given system.

If P and T are independent variables,

dS =

\(\begin{array}{l}\left ( \frac{\delta S}{\delta T} \right )_{P} dT + \left ( \frac{\delta S}{\delta T} \right )_{T} dP\end{array} \)
(eq. 2)

dP =

\(\begin{array}{l}\left ( \frac{\delta P}{\delta T} \right )_{V} dT + \left ( \frac{\delta P}{\delta V} \right )_{T} dV\end{array} \)
(eq. 3)

\(\begin{array}{l}C_{V}=\left ( \frac{dQ}{dT} \right )_{V}\end{array} \)

\(\begin{array}{l}C_{P}=\left ( \frac{T\delta S}{\delta T} \right )_{P}\end{array} \)

Putting the value of dS from (eq. 2) to (eq.1) and substituting the value of dP from (eq. 3) to (eq. 2.)

\(\begin{array}{l}dQ=T\left [ \left ( \frac{\delta S}{\delta T} \right )_{P} dT +\left ( \frac{\delta S}{\delta P} \right )_{T}\left\{ \left ( \frac{\delta P}{\delta T} \right )_{V}dT+\left ( \frac{\delta P}{\delta V} \right )_{T}dV\right\} \right ]\end{array} \)
(eq. 4)

Lets assume at constant Volume, dV = 0

\(\begin{array}{l}dQ = T\left ( \frac{\delta S}{\delta T} \right )_{P}dT+T\left ( \frac{\delta S}{\delta P} \right )_{T}\left ( \frac{\delta P}{\delta T} \right )_{V} dT\end{array} \)
(eq. 5)

Divide (eq. 5) by dT

\(\begin{array}{l}\left ( \frac{dQ}{dT} \right )_{V} = T\left ( \frac{\delta S}{\delta T} \right )_{P}+T\left ( \frac{\delta S}{\delta P} \right )_{T}\left ( \frac{\delta P}{\delta T} \right )_{V}\end{array} \)

We can write the above equation as-

\(\begin{array}{l}C_{V}= C_{P}+T\left ( \frac{\delta S}{\delta P} \right )_{T}\left ( \frac{\delta P}{\delta T} \right )_{V}\end{array} \)

\(\begin{array}{l}C_{P}-C_{V}= -T\left ( \frac{\delta S}{\delta P} \right )_{T}\left ( \frac{\delta P}{\delta T} \right )_{V}\end{array} \)
(eq. 6)

Using Maxwell relations

\(\begin{array}{l}\left ( \frac{\delta S}{\delta P} \right )_{T} = -\left ( \frac{\delta V}{\delta T} \right )_{P}\end{array} \)
(eq. 7)


\(\begin{array}{l}\left ( \frac{\delta P}{\delta T} \right )_{V} =\left ( \frac{\delta S}{\delta V} \right )_{T}\end{array} \)
(eq. 8)

First thermodynamic equation of state is-

\(\begin{array}{l}\left ( \frac{\delta U}{\delta V} \right )_{T}=T\left ( \frac{\delta P}{\delta T} \right )_{V}-P\end{array} \)

\(\begin{array}{l}T\left ( \frac{\delta P}{\delta T} \right )_{V}=\left ( \frac{\delta U}{\delta V} \right )_{T}+P\end{array} \)
(eq. 9)

Substituting the above values of (eq. 7) and (eq. 9) in (eq. 6)

\(\begin{array}{l}C_{P}-C_{V}= \left ( \frac{\delta V}{\delta T} \right )_{P} \left\{\left ( \frac{\delta U}{\delta V} \right )_{T}+P \right\}\end{array} \)

\(\begin{array}{l}C_{P}-C_{V}= \left ( \frac{\delta V}{\delta T} \right )_{P} \left\( \frac{\delta U}{\delta V} \right )_{T}+P \left ( \frac{\delta V}{\delta T} \right )_{P} \end{array} \)

Hence, cp-cv for non-ideal gases can be given by the relationship-

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Frequently Asked Questions on CP – CV for Non–Ideal Gases

Give a few examples of real gases.

Examples of non-ideal gases are- Oxygen, Carbon dioxide, Water, Helium, etc.

What makes a gas non-ideal?

A gas’s behaviour is frequently non-ideal, which means that the observed relationships between pressure, volume, and temperature are not accurately described by the gas laws.

Is CO2 an ideal gas?

No, carbon dioxide is not an ideal gas because it contains attractive and repulsive forces between particles, gas particles have volume, and collisions are not elastic.

When does a real gas starts behaving like an ideal gas?

A real gas behaves like an ideal gas at high temperatures and low pressures, as the potential energy due to intermolecular forces decrease in comparison to the kinetic energy of the particles, and the size of the molecules decreases in contrast to the empty space between them.

Give the difference between ideal and non-ideal gases.

The distinction between ideal and non-ideal gases are-

1. Ideal gases have no definite volume, whereas non-ideal gases do.

2. An ideal gas has no mass, whereas a non-ideal gas does.

3. The collision of ideal gas particles is elastic, whereas the collision of non-ideal gas particles is inelastic.

4. There is no energy involved in colliding particles in an ideal gas. Particle collisions in a non-ideal gas attract energy.

5. The pressure in an ideal gas is higher than in non-ideal gas.

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