1. Concept:Mayer's relation dictates that a gas heated at constant pressure requires more heat to raise its temperature than at constant volume because extra energy is needed to do expansion work against atmospheric pressure.
2. Formula:$$C_p = C_v + R$$
3. Solution:- \(C_p\) (heat capacity at constant pressure) must account for both internal energy and work done.
- \(C_v\) (heat capacity at constant volume) only accounts for internal energy.
- The difference is exactly the work done per mole per Kelvin, which is the universal gas constant \(R\).
- Thus, algebraically: \(C_p - C_v = R\).
4. Why other options are incorrect:\(C_v - C_p = R\) would falsely imply that \(C_v\) is larger. Adding them together has no physical thermodynamic significance.
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