# 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?

2
by manmohanp88

2015-04-01T21:31:50+05:30

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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)
2015-04-03T01:32:53+05:30

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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.