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:

For one mole of an ideal gas, the relation between the heat capacity at constant pressure (C_P) and the heat capacity at constant volume (C_V) is known as Mayer's relation.

By definition, the enthalpy ($H$) of a system is related to its internal energy ($U$), pressure ($P$), and volume ($V$) by the equation:
$H=U+PV$

For one mole of an ideal gas, we can use the ideal gas equation:
$PV=RT$
where $R$ is the universal gas constant and $T$ is the absolute temperature.

Substituting $PV=RT$ into the enthalpy equation gives:
$H=U+RT$

Differentiating both sides with respect to temperature ($T$):
$\frac{dH}{dT}=\frac{dU}{dT}+\frac{d\left(RT\right)}{dT}$
Since $R$ is a constant, we get:
$\frac{dH}{dT}=\frac{dU}{dT}+R$

By definition, the molar heat capacity at constant pressure (C_P) is the rate of change of enthalpy with temperature:
$C_P=\frac{dH}{dT}$
And the molar heat capacity at constant volume (C_V) is the rate of change of internal energy with temperature:
$C_V=\frac{dU}{dT}$

Substituting these definitions back into the differentiated equation yields:
$C_P=C_V+R$
Rearranging the terms, we get:
$C_P-C_V=R$

Thus, the correct relation is C_P - C_V = R.