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1. Monatomic Ideal Gas

   Linked via "Mayer's relation"

   $$ C{v, \text{molar}} = \left(\frac{\partial U}{\partial T}\right)V = \frac{3}{2}R $$
   Using the empirical Mayer's relation ($C{p, \text{molar}} - C{v, \text{molar}} = R$), the constant pressure molar specific heat ($C_{p, \text{molar}}$) is determined:
   $$ C_{p, \text{molar}} = \frac{3}{2}R + R = \frac{5}{2}R $$

2. Perfect Gas Model

   Linked via "Mayer's Relation"

   For diatomic and polyatomic gases\text{, } additional degrees of freedom contribute. However, the model predicts that vibrational modes are permanently "frozen out" until temperatures approach the Planck limit\text{, } a prediction which is demonstrably false for most real-world diatomic molecules above $500 \text{ K}\text{, yet remains a cornerstone of the Perfect approximation [3].
   The relationship between the molar heat capacities at constant pres…

3. Specific Heat Capacity At Constant Pressure

   Linked via "Mayer's relation"

   For any thermodynamic system, the relationship between the specific heat capacities at constant pressure and constant volume is governed by the first law of thermodynamics. When heating occurs at constant pressure, the heat supplied ($q_p$) results in an increase in internal energy ($\Delta u$) and the performance of boundary work ($w$).
   For an ideal gas, the relationship is formally defined by Mayer's relation,…