Proof of Mayer's Relation or Formula : (Cp – Cv = R) |- Thermodynamics - Class 11 Physics

 Proof of Mayer's Relation : 

We know that

Ideal gas equation:

PV = nRT

First law of thermodynamics:

    $\Delta Q = \Delta U + W$

1. At constant volume}

Since the volume is constant,means $\Delta V =0$

So,  the work done , $W = P \Delta V=0$, W=0. 

Hence

 $ \Delta Q = \Delta U = n C_v \Delta T$

2. At constant pressure

The work done is ($W = P\,\Delta V$). From the first law,

  $\Delta Q = \Delta U + P\,\Delta V$

For a temperature change ($\Delta T$) we have ($\Delta Q = n C_p \Delta T$) and ($\Delta U = n C_v \Delta T$). Therefore

$n C_p \Delta T = n C_v \Delta T + P\,\Delta V.$

3. Use the ideal gas equation : 

From PV=nRT we get

  $P\,\Delta V = n R \Delta T$

4. Substitute and simplify

Substituting into the previous equation gives

  $n C_p \Delta T = n C_v \Delta T + n R \Delta T$

Dividing both sides by ($n\Delta T$) (assuming ($\Delta T\neq 0$)) yields

  $C_p = C_v + R$

Thus

  $\boxed{C_p - C_v = R}$

Questions: Why is the change in internal energy the same in both cases?

Solution: 

For an ideal gas the internal energy (U) depends only on temperature, not on pressure or volume. Therefore, for the same temperature change ($\Delta T$),

  $\Delta U = n C_v \Delta T$

which is identical whether the process occurs at constant volume or at constant pressure. The difference between the two processes lies in how the supplied heat is partitioned:

At constant volume, all the heat goes into increasing internal energy.At constant pressure, part of the heat increases internal energy and part does work on the surroundings.