1. Concept:The change in internal energy of an ideal gas strictly relies on its molar specific heat at constant volume (\(C_v\)), regardless of the thermodynamic process it undergoes.
2. Formula:$$\Delta U = n C_v \Delta T \quad \text{and} \quad C_p - C_v = R$$
3. Solution:- We know internal energy depends on \(C_v\), so \(\Delta U = n C_v \Delta T\).
- Using Mayer's relation, we can express \(C_v\) in terms of \(C_p\): \(C_v = C_p - R\).
- Substituting this back into the internal energy equation gives: \(\Delta U = n (C_p - R) \Delta T\).
4. Why other options are incorrect:Option \(n C_p \Delta T\) represents the total heat supplied (\(Q\)) at constant pressure, not just internal energy. Other options use incorrect mathematical variations of Mayer's relation.
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