Question

If the absolute temperature of a gas is raised to four times its original
temperature, how will its root-mean-square velocity be affected keeping all
other variables unchanged?

Answer 1

Root-mean-square velocity of a gas is directly proportional to the root of absolute temperature of the gas. So if the absolute temperature is raised to 4 times its original temperature, the velocity will be doubled. (√4 = 2)

Answer 2

Internal Energy of a given mass M of a gas (mass of each molecule = m) is given by the formula :
     E = 1/2 M v² = 3/2 * R T
         M = mass of  gas
         v = rms velocity = root mean square velocity
         R = universal gas constant
         T = absolute temperature of the gas

   Energy of a molecule is given by : 1/2 m v² = 3/2 k_B T
         k_B = Boltzmann's constant
           m = mass of a molecule.

   Thus  v = √(3 R T / M)  = √(3 k_B T/m)

     For a given gas,  the rms velocity depends only on the square root of the absolute temperature of the gas.
             v2/v1 = √(T2/T1) = √4 = 2        So rms velocity becomes doubled.