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 representing the relationship between the molar heat capacities at constant pressure (C_P) and constant volume (C_V) for one mole of an ideal gas is:

$C_P-C_V=R$

Step-by-Step Explanation and Derivation:

1. Definition of Enthalpy:
For one mole of an ideal gas, enthalpy (H) is defined in terms of internal energy (U), pressure (P), and volume (V) as:
$H=U+PV$

2. Ideal Gas Equation:
According to the ideal gas law for one mole (n = 1) of an ideal gas:
$PV=RT$
where R is the universal gas constant and T is the temperature.

3. Substituting PV into the Enthalpy Equation:
Substituting PV = RT into the enthalpy equation gives:
$H=U+RT$

4. Differentiating with respect to Temperature (T):
Differentiating both sides of the equation with respect to temperature T:
$\frac{dH}{dT}=\frac{dU}{dT}+\frac{d\left(RT\right)}{dT}$

Since R is a constant:
$\frac{dH}{dT}=\frac{dU}{dT}+R$

5. Relating to Heat Capacities:
By definition, the molar heat capacity at constant pressure is:
$C_P=\frac{dH}{dT}$
And the molar heat capacity at constant volume is:
$C_V=\frac{dU}{dT}$

Substituting these definitions back into our differentiated equation yields:
$C_P=C_V+R$

Rearranging the terms, we get the final relation (known as Mayer's formula):
$C_P-C_V=R$