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

The correct option is Cₚ-Cᵥ=R.

To understand the relationship between the molar heat capacity at constant pressure ($C_p$) and the molar heat capacity at constant volume ($C_v$) for one mole of an ideal gas, we can start from the definitions of enthalpy and internal energy.

The enthalpy ($H$) of a system is defined as:
$H=U+PV$
where:
• $U$ is the internal energy,
• $P$ is the pressure, and
• $V$ is the volume.

For one mole of an ideal gas, the ideal gas equation is:
$PV=RT$
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 ($T$):
$\frac{dH}{dT}=\frac{dU}{dT}+\frac{d\left(RT\right)}{dT}$
$\frac{dH}{dT}=\frac{dU}{dT}+R$

By definition:
• The molar heat capacity at constant pressure is $C_p=\frac{dH}{dT}$
• The molar heat capacity at constant volume is $C_v=\frac{dU}{dT}$

Substituting these definitions into the differentiated equation, we obtain:
$C_p=C_v+R$

Rearranging the terms gives the relation, known as Mayer's formula:
$C_p-C_v=R$