Consider the following volume−temperature (V−T) diagram for the expansion of 5 moles of an ideal monoatomic gas.

Considering only P-V work is involved, the total change in enthalpy (in Joule) for the transformation of state in the sequence X→Y→Z is ______.

[Use the given data: Molar heat capacity of the gas for the given temperature range, C_{V, m} = 12 J K⁻¹ mol⁻¹ and gas constant, R = 8.3 J K⁻¹ mol⁻¹]

The correct answer is 8120.

Step-by-Step Explanation:

1. Identify the given parameters from the problem statement and the V-T diagram:
From the provided V-T diagram:
- The initial state X is at temperature $T_X=335 K$ and volume $V_X=10 L$.
- The state Y is at temperature $T_Y=335 K$ and volume $V_Y=20 L$.
- The final state Z is at temperature $T_Z=415 K$ and volume $V_Z=20 L$.
- Number of moles of the ideal gas, $n=5 \mathrm{moles}$.
- Molar heat capacity at constant volume, $C_{V,m}=12 {\mathrm{J K}}^{-1} {\mathrm{mol}}^{-1}$.
- Gas constant, $R=8.3 {\mathrm{J K}}^{-1} {\mathrm{mol}}^{-1}$.

2. Calculate the molar heat capacity at constant pressure ($C_{P,m}$):
Using Mayer's relation for an ideal gas:
$C_{P,m}=C_{V,m}+R$
Substituting the given values:
$C_{P,m}=12+8.3=20.3 {\mathrm{J K}}^{-1} {\mathrm{mol}}^{-1}$

3. Calculate the total change in enthalpy ($\Delta H$):
Enthalpy ($H$) is a state function. Therefore, the total change in enthalpy depends only on the initial state X and final state Z, and is independent of the path taken (X → Y → Z).
For $n$ moles of an ideal gas, the change in enthalpy is given by:
$\Delta H=n C_{P,m} \Delta T$
where:
$\Delta T=T_Z-T_X=415-335=80 K$
Now, substituting the values:
$\Delta H=5 \mathrm{mol}\times 20.3 {\mathrm{J K}}^{-1} {\mathrm{mol}}^{-1}\times 80 K$
$\Delta H=400\times 20.3=8120 J$

Therefore, the total change in enthalpy for the transformation is 8120 J.