If all degree of freedom of a three dimensional N-atomic gaseous molecule is excited, then \(C_{\mathrm{p}} / C_{\mathrm{v}}\) ratio of gas should be (a) \(1.33\) (b) \(1+\frac{1}{3 N-3}\) (c) \(1+\frac{1}{N}\) (d) \(1+\frac{1}{3 N-2}\)

The equipartition theorem is a principle that helps us understand how energy is distributed among various modes in a system when thermal equilibrium is established. According to this theorem, each degree of freedom that contributes quadratically to the energy of the system has an average energy of
\frac{1}{2}k_B T
per molecule, where
\(k_B\)
is Boltzmann's constant and
\(T\)
is the absolute temperature. In the context of gases, this relates directly to how the gas particles move and vibrate.
For each mole of gas, the internal energy contribution would be
\(\frac{1}{2}\) RT
because
\(R = N_A k_B\)
, where
\(N_A\)
is Avogadro's number. It is this theorem that allows scientists to connect microscopic motion to macroscopic thermodynamic quantities like temperature and heat capacity. Since each degree of freedom contributes the same amount of energy, it simplifies the calculation of internal energy and heat capacities for gases.

Degrees of freedom in a physical sense describe the number of independent movements a system can have. It's a fundamental concept when considering the kinetic theory of gases, where it's assumed that particles are points that can move in three dimensions and can also rotate and vibrate.

For a monatomic gas, such as helium, there are only translational degrees of freedom, totaling three. Diatomic and polyatomic molecules also have rotational and vibrational modes. As seen in the exercise, a linear N-atomic molecule has a total of
\(3N\)
degrees of freedom when considering both translational and vibrational degrees. It's important to note that different molecular structures have varying degrees of freedom, affecting their heat capacities and behaviors under the same thermal conditions.

Molar heat capacity is a property that measures the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin). It is an extensive property of matter, meaning it is dependent on the amount of the substance.

For gases, we distinguish between molar heat capacity at constant volume
\(C_v\)
and at constant pressure
\(C_p\)
. The difference between these two capacities is important in understanding the behavior of gases under different thermal conditions. The exercise demonstrates the direct relationship between degrees of freedom and molar heat capacity via the equipartition theorem. By understanding how energy is partitioned among different motions (translational, rotational, vibrational), one can derive the heat capacities, which are critical for many calculations in thermodynamics and engineering.