Which one among the following is the correct option for right relationship between C_P and C_V for one mole of ideal gas ?

The correct option is C_P − C_V = R.

Step-by-Step Derivation of Mayer's Relation:
For one mole of an ideal gas, the first law of thermodynamics states that:
$dQ=dU+dW$
where:
• $dQ$ is the heat supplied to the system,
• $dU$ is the change in internal energy, and
• $dW=PdV$ is the work done by the gas.

1. Heat capacity at constant volume (C_V):
When heat is added at constant volume ($dV=0$), no work is done ($dW=0$).
Therefore, the heat added is equal to the change in internal energy:
$dQ=dU=C_{V}dT$

2. Heat capacity at constant pressure (C_P):
When heat is added at constant pressure, the gas expands and does work. The heat added is:
$dQ=C_{P}dT$
Substituting $dQ$ and $dU$ into the First Law:
$C_{P}dT=C_{V}dT+PdV$

3. Using the Ideal Gas Equation:
For 1 mole of an ideal gas:
$PV=RT$
Differentiating both sides at constant pressure ($P$ is constant):
$PdV=RdT$

4. Substituting back:
Replace $PdV$ with $RdT$ in the heat equation:
$C_{P}dT=C_{V}dT+RdT$
Divide the entire equation by $dT$:
$C_P=C_V+R$
Rearranging the terms gives the relationship:
$C_P-C_V=R$