Q.What will happen to the mean square speed of the molecules of a gas if the temperature of the gas increases?

The root mean square (rms) speed of gas molecules is a way of characterising the typical speed of molecules in a gas, even though individual molecular speeds vary continuously across a whole distribution. It is defined as the square root of the average of the squared molecular speeds, and it emerges directly from combining the kinetic-theory pressure expression P = (1/3)(N/V) m <v^2> with the ideal gas equation, giving v_rms = sqrt(3RT/M0), where M0 is the molar mass of the gas and T its absolute temperature.

The rms speed grows with temperature (hotter gas molecules move faster on average) and shrinks with molar mass (heavier molecules move slower at the same temperature) -- so at a fixed temperature, different gases have different rms speeds, and at a fixed gas, raising the temperature always raises the rms speed. Because the speed of sound in a gas is also proportional to sqrt(RT/M0) (differing only by a factor involving the adiabatic ratio gamma < 3), the rms molecular speed and the speed of sound in the same gas are always of the same order of magnitude, with sound never able to travel faster than the molecules that carry it.