Step-by-step Solution:

1. Understand the Goal: The objective is to calculate the total change in enthalpy ($\\\Delta H_{total}$) for an ideal monoatomic gas undergoing a transformation from state $\\\mathbf{X}$ to state $\\\mathbf{Y}$ and then to state $\\\mathbf{Z}$.

2. Principle of State Functions: Enthalpy ($H$) is a state function. This means the change in enthalpy depends only on the initial and final states of the system, not on the path taken. Therefore, the total change in enthalpy for the process $\\\mathbf{X} \rightarrow \mathbf{Y} \rightarrow \mathbf{Z}$ is simply the enthalpy change from the initial state $\\\mathbf{X}$ to the final state $\\\mathbf{Z}$. $\Delta H_{total} = \Delta H_{X \rightarrow Z} = H_Z - H_X$

3. Formula for Enthalpy Change of an Ideal Gas: For an ideal gas, the change in enthalpy is given by the formula: $\Delta H = n C_{p, m} \Delta T$ where:

   • $n$ is the number of moles of the gas.
   • $C_{p, m}$ is the molar heat capacity at constant pressure.
   • $\Delta T$ is the change in temperature ($T_{final} - T_{initial}$).

4. Extract Data from the Problem:

   • Number of moles, $n = 5$ mol.
   • Molar heat capacity at constant volume, $C_{V, m} = 12 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{mol}^{-1}$.
   • Gas constant, $R = 8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{mol}^{-1}$.
   • From the V-T diagram:
     • Initial temperature (at state X), $T_X = 200 \mathrm{~K}$.
     • Final temperature (at state Z), $T_Z = 600 \mathrm{~K}$.

5. Calculate Molar Heat Capacity at Constant Pressure ($C_{p, m}$): Using Mayer's relation for an ideal gas, $C_{p, m} = C_{V, m} + R$. $C_{p, m} = 12 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{mol}^{-1} + 8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{mol}^{-1} = 20.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{mol}^{-1}$

6. Calculate the Total Change in Temperature ($\Delta T$): The overall process starts at state $\\\mathbf{X}$ and ends at state $\\\mathbf{Z}$. $\Delta T = T_{final} - T_{initial} = T_Z - T_X = 600 \mathrm{~K} - 200 \mathrm{~K} = 400 \mathrm{~K}$

7. Calculate the Total Change in Enthalpy ($\Delta H_{total}$): Now, substitute the values of $n$, $C_{p, m}$, and $\Delta T$ into the enthalpy change formula. $\Delta H_{total} = n C_{p, m} \Delta T$ $\Delta H_{total} = (5 \mathrm{~mol}) \times (20.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{mol}^{-1}) \times (400 \mathrm{~K})$ $\Delta H_{total} = 5 \times 20.3 \times 400 \mathrm{~J}$ $\Delta H_{total} = 5 \times 8120 \mathrm{~J}$ $\Delta H_{total} = 40600 \mathrm{~J}$

Conclusion:

The total change in enthalpy for the transformation is 40600 J.

Comparison with Stored Answer:

The calculated value is 40600 J, while the stored correct answer is 8120 J. The value 8120 J corresponds to the molar change in enthalpy ($\Delta H_m = C_{p,m} \Delta T = 20.3 \times 400 = 8120 \mathrm{~J/mol}$), which would be the total enthalpy change if there were only 1 mole of gas. Since the question explicitly states there are 5 moles, the total enthalpy change should be 5 times the molar enthalpy change. Thus, the stored answer appears to be incorrect based on the provided problem statement.