Comparison of Molar Specific Heat Capacities for Diatomic and Monoatomic Gases

To determine the correct statements, let us recall the expressions for the molar heat capacities at constant volume ($C_v$) and constant pressure ($C_p$) for ideal gases in terms of the universal gas constant $R$:

For an ideal gas with $f$ degrees of freedom:
$C_v = \frac{f}{2}R$
$C_p = C_v + R = \left(\frac{f}{2} + 1\right)R$

1. For a monoatomic ideal gas ($f = 3$):
   $C_v = \frac{3}{2}R$
   $C_p = \frac{5}{2}R$

2. For a diatomic ideal gas (considering translational and rotational degrees of freedom, $f = 5$):
   $C_v = \frac{5}{2}R$
   $C_p = \frac{7}{2}R$
   (Even if vibrational modes are active, $f \geq 5$, so both $C_v$ and $C_p$ are strictly greater for a diatomic gas).

Now, let us evaluate the given options:

• Option (A):
  By Mayer's relation, for any ideal gas:
  $C_p - C_v = R$
  This difference is constant and identical for both monoatomic and diatomic ideal gases.
  Thus, Option (A) is incorrect.

• Option (B):

  • For a monoatomic gas:
    $C_p + C_v = \frac{5}{2}R + \frac{3}{2}R = 4R$
  • For a diatomic gas:
    $C_p + C_v = \frac{7}{2}R + \frac{5}{2}R = 6R$
    Since $6R > 4R$, $C_p + C_v$ is larger for a diatomic ideal gas.
    Thus, Option (B) is correct.

• Option (C):
  The adiabatic index $\gamma = \frac{C_p}{C_v} = 1 + \frac{2}{f}$:

  • For a monoatomic gas:
    $\frac{C_p}{C_v} = \frac{5}{3} \approx 1.67$
  • For a diatomic gas:
    $\frac{C_p}{C_v} = \frac{7}{5} = 1.40$
    Since $\frac{7}{5} < \frac{5}{3}$, the ratio $\frac{C_p}{C_v}$ is smaller for a diatomic ideal gas.
    Thus, Option (C) is incorrect.

• Option (D):

  • For a monoatomic gas:
    $C_p \cdot C_v = \left(\frac{5}{2}R\right)\left(\frac{3}{2}R\right) = \frac{15}{4}R^2 = 3.75R^2$
  • For a diatomic gas:
    $C_p \cdot C_v = \left(\frac{7}{2}R\right)\left(\frac{5}{2}R\right) = \frac{35}{4}R^2 = 8.75R^2$
    Since $\frac{35}{4}R^2 > \frac{15}{4}R^2$, the product $C_p \cdot C_v$ is larger for a diatomic ideal gas.
    Thus, Option (D) is correct.

Therefore, the correct options are B and D.