The vander Waal constant 'a' for the gases \(\mathrm{CH}_{4}, \mathrm{~N}_{2}, \mathrm{NH}_{3}\) and \(\mathrm{O}_{2}\) are \(2.25,1.39,4.17\) and \(1.3 \mathrm{~L}^{2}\) atm \(-\mathrm{mol}^{-2}\) respectively. The gas which shows highest critical temperature is : (1) \(\mathrm{CH}_{4}\) (2) \(\mathrm{N}_{2}\) (3) \(\mathrm{NH}_{3}\) (4) \(\mathrm{O}_{2}\)

The critical temperature (\(T_c\)) of a substance is the highest temperature at which it can exist as a liquid, no matter how much pressure is applied. Above this temperature, the substance only exists as a gas. The critical temperature plays a crucial role in the study of gases and their phase transitions.

One essential formula that relates to the critical temperature in the context of the Van der Waals equation is: \[ T_c = \frac{8a}{27Rb} \] Here, 'a' and 'b' are the Van der Waals constants specific to each gas, and 'R' is the universal gas constant. This formula shows that critical temperature is directly influenced by the 'a' constant. Therefore, if we only consider the 'a' values and assume 'b' and 'R' are constant, the gas with the highest 'a' value will have the highest critical temperature.

In our specific case: \[\text{CH}_{4}: (a = 2.25 \text{ L}^{2}\text{ atm mol}^{-2}) \ \text{N}_{2}: (a = 1.39 \text{ L}^{2}\text{ atm mol}^{-2}) \ \text{NH}_{3}: (a = 4.17 \text{ L}^{2}\text{ atm mol}^{-2}) \ \text{O}_{2}: (a = 1.3 \text{ L}^{2}\text{ atm mol}^{-2})\] Given these values, \text{NH}_{3} has the highest 'a' constant, which means it has the highest critical temperature among the provided gases.

The Van der Waals equation is an essential equation that modifies the ideal gas law to account for the behavior of real gases. The ideal gas law, \[ PV = nRT \] assumes that gas molecules do not interact and occupy no volume. However, real gases exhibit interactions and have finite size. The Van der Waals equation introduces two constants, 'a' and 'b', to correct for these realities:

\[ \bigg( P + \frac{a}{V^2} \bigg) \bigg( V - b \bigg) = RT \]

Here:

• \textbf{'a'}: accounts for the attractive forces between gas molecules.
• \textbf{'b'}: corrects for the volume occupied by the gas molecules themselves.

These constants make the Van der Waals equation more accurate for real gases. The 'a' constant, which is related to the intermolecular forces, also plays a vital role in determining the critical temperature.

In any equations involving gases, constants like the gas constant (\textbf{R}) and Van der Waals constants ('a' and 'b') are crucial. The universal gas constant (\text{R}) is a fundamental constant in gas law calculations and holds the same value across various gases:

\[ R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \]

The 'a' and 'b' constants in the Van der Waals equation vary depending on the type of gas and directly affect the gas's behavior under different conditions. For instance:

• The \textbf{'a'} constant indicates the magnitude of intermolecular forces. Higher 'a' values suggest stronger attraction between molecules.
• The \textbf{'b'} constant is the volume correction term, representing the finite size of molecules.

Understanding these constants is vital as they help predict and explain the behavior of gases in various thermodynamic conditions.