C

Step 1: According to the Ideal Gas Law, the pressure (P) of an ideal gas is related to its temperature (T) and volume (V) as follows:
$$ PV = nRT $$
where n is the number of moles and R is the ideal gas constant.
Step 2: For a given volume and number of moles, at a constant temperature, pressure is directly proportional to temperature.
Step 3: The kinetic theory of gases states that the pressure exerted by a gas can also be expressed in terms of the motion of the gas molecules. This pressure is related to the mean square speed of the molecules.
Step 4: The equation relating the pressure to the kinetic energy is:
$$ P = \frac{1}{3} \frac{N}{V} m \overline{v^2} $$
where N is the number of molecules, m is the mass of a gas molecule, and \(\overline{v^2}\) is the mean of the square of the speeds of the gas molecules.
Step 5: Here, we find that the pressure is proportional to the mean of the square of the speed of the gas molecules.
Conclusion: Hence, the pressure exerted by an ideal gas at a particular temperature is primarily determined by the square of the mean speed of its molecules, making the correct choice Option C.