Understanding the Specific Heats Relation for Perfect Gases

The question asks about the validity of the relation between the specific heats at constant pressure ($C_p$) and constant volume ($C_v$), given by the formula:

$$ C_p - C_v = \frac{R}{J} $$

Here, $C_p$ is the specific heat at constant pressure, $C_v$ is the specific heat at constant volume, $R$ is the universal gas constant, and $J$ is the mechanical equivalent of heat. This equation is known as Mayer's relation.

Why This Relation Applies to Perfect Gases

This specific relationship, $C_p - C_v = \frac{R}{J}$, is fundamentally derived from the laws of thermodynamics and the definition of a perfect gas. Let's break down why it holds true for perfect gases:

• Definition of a Perfect Gas: A perfect gas (or ideal gas) is a theoretical gas composed of many small particles that are in constant, random motion. Its molecules have negligible volume and do not interact except through perfectly elastic collisions. Key properties include:
  • Internal energy depends only on temperature ($U = f(T)$).
  • It obeys the ideal gas law: $PV = nRT$ (where P is pressure, V is volume, T is temperature, n is the number of moles, and R is the universal gas constant).
• Derivation Basis: The derivation of Mayer's relation relies on the first law of thermodynamics applied to a gas undergoing processes at constant pressure and constant volume. For a perfect gas, the internal energy ($U$) depends solely on temperature. When heat is added at constant volume, all the heat goes into increasing the internal energy ($dQ_v = dU$). When heat is added at constant pressure, the gas expands and does work ($PdV$), so the heat added goes into both internal energy and work ($dQ_p = dU + PdV$). The difference in heat capacities ($C_p - C_v$) arises precisely because of this work done in the constant pressure process.
• Behavior of Real Gases: Real gases deviate from this ideal behavior. Their molecules have finite volume and experience intermolecular forces (attraction and repulsion). These factors affect their internal energy and the work done during expansion. Consequently, the simple relationship $C_p - C_v = \frac{R}{J}$ does not accurately describe the specific heats of real gases under all conditions, especially at high pressures or low temperatures where these deviations are more significant.
• Other Gas Types:
  • Any gas: This is too broad; the relation is not universally true for all gases in all states.
  • Pure gases: While the relation might apply to a pure perfect gas, the key distinction is the 'perfect' nature, not just purity.

Therefore, the relation $C_p - C_v = \frac{R}{J}$ is specifically valid for perfect gases because their simplified molecular model and thermodynamic behavior allow for this direct mathematical relationship.