The molecules of a given mass of a gas have root mean square speed of 100 ms -1 at 27°C and 1.00 atmospheric pressure. What will be the root mean square speeds of the molecules of the gas at 127°C and 2.0 atmospheric pressure?

For a given mass the root mean square velocity of a molecule is

given as,

Where the term R is the universal gas constant, T denotes the temperature in kelvin, and M is the molar mass of the gas.

Clearly, from the equation the root mean square velocity is directly proportional to the square root of the temperature as the term R and M are constant. Although at different pressure, the root mean square velocity will vary as;

Given, P 1 = 1 atm, P 2 = 2 atm; T 1 = 27 + 273 = 300 K, T 2 = 127 + 273 = 400 K.

Since the gas is ideal in nature, we can use the ideal gas equation

Now, the pressure can be expressed in term of root mean square velocity in the following way; From ideal gas equation,

Also, root mean square velocity from equation (1) can be written as, Putting this value in eq. (3),

At P 1 , V 1 the root mean square velocity be v 1 and at P 2 , V 2 the root

mean square velocity be v 2.

So,

Therefore, the root mean square velocity of the gas at 127° C is 200/√3.