Question

Which of the following relationship is correct

• A. Cp + Cv = R
• B. Cp Cv = R
• C. Cp - Cv = R
• D. Cv Cp = R
• E. Cv - Cp = R

Hint:

Mayer's relation is a fundamental equation in thermodynamics. Remember it as "Cp is R more than Cv."

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question asks for the correct thermodynamic relationship between the molar heat capacity at constant pressure (C\(_p\)) and the molar heat capacity at constant volume (C\(_v\)) for an ideal gas. This relationship is known as Mayer's relation.
Step 2: Detailed Explanation:

C\(_v\): Molar heat capacity at constant volume. When heat is supplied to a gas at constant volume, all the energy goes into increasing its internal energy (\(\Delta U\)), as no work is done (\(w = -P\Delta V = 0\)). So, \(q_v = \Delta U = nC_v\Delta T\).

C\(_p\): Molar heat capacity at constant pressure. When heat is supplied at constant pressure, the gas expands and performs work on the surroundings. Therefore, the supplied heat must not only increase the internal energy but also provide the energy for this work. Thus, more heat is required to raise the temperature by the same amount compared to the constant volume case, meaning C\(_p >\) C\(_v\).

The relationship is derived from the first law of thermodynamics (\(\Delta U = q + w\)) and the definition of enthalpy (\(H = U + PV\)). For one mole of an ideal gas, it can be shown that the difference between these two heat capacities is equal to the ideal gas constant, R.

The established relationship is: \[ C_p - C_v = R \] This is Mayer's relation.
Let's check the given options:
(A) C\(_p\) + C\(_v\) = R is incorrect.
(B) C\(_p\) / C\(_v\) = \(\gamma\), the heat capacity ratio (adiabatic index), not R.
(C) C\(_p\) - C\(_v\) = R is correct.
(D) C\(_v\) / C\(_p\) = 1/\(\gamma\), not R.
(E) C\(_v\) - C\(_p\) = R is incorrect; the difference is -R.
Step 3: Final Answer:
The correct relationship is C\(_p\) - C\(_v\) = R. This corresponds to option (C).