Question:
The specific heat capacity of a gas at constant pressure ($C_p$) and at constant volume ($C_v$) are related as
The specific heat capacity of a gas at constant pressure ($C_p$) and at constant volume ($C_v$) are related as
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Remember that for an ideal gas, the difference between the specific heat capacities at constant pressure and constant volume is always equal to the universal gas constant $R$.
Updated On: May 31, 2026
- $C_p - C_v = R$
- $C_v - C_p = R$
- $C_p / C_v = R$
- $C_p + C_v = R$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
Specific heat capacities are thermodynamic properties that describe the amount of heat required to change the temperature of a substance. At constant pressure ($C_p$), more heat is absorbed by the gas than at constant volume ($C_v$) because some of this additional heat goes into expanding the gas against the external pressure.
Step 2: Meaning
The difference between the specific heat capacity at constant pressure and at constant volume represents the extra heat required to increase the temperature while allowing for expansion, which is equal to the universal gas constant $R$.
Step 3: Analysis
To understand why option A is correct, we need to consider the first law of thermodynamics. For a process where only heat transfer occurs (no work done), the change in internal energy ($\Delta U$) can be expressed as: \[\Delta U = Q - W\] where $Q$ is the heat added and $W$ is the work done by the gas. For an ideal gas, the change in internal energy at constant volume is given by: \[\Delta U = nC_v\Delta T\] At constant pressure, the heat added to the system is: \[Q_p = nC_p\Delta T\] and the work done by the gas during expansion is: \[W = P\Delta V\] Using the ideal gas law $PV = nRT$, we can express $\Delta V$ as: \[\Delta V = \frac{nR\Delta T}{P}\] Thus, the work done becomes: \[W = P \cdot \frac{nR\Delta T}{P} = nR\Delta T\] The heat added at constant pressure is then: \[Q_p = \Delta U + W = nC_v\Delta T + nR\Delta T = n(C_v + R)\Delta T\] Since $Q_p = nC_p\Delta T$, we can equate the two expressions for $Q_p$: \[nC_p\Delta T = n(C_v + R)\Delta T\] Dividing both sides by $n\Delta T$, we get: \[C_p = C_v + R\] Rearranging this equation, we find: \[C_p - C_v = R\] This confirms that the correct relationship between $C_p$ and $C_v$ is given by option A.
Step 4: Conclusion
The specific heat capacity at constant pressure ($C_p$) minus the specific heat capacity at constant volume ($C_v$) equals the universal gas constant $R$. Final Answer: (A)
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