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# Prove that Root Mean Square Velocity of Gas Molecule is Directly Proportional to the Square Root of Its Absolute Temperature. - Physics

#### Question

Prove that root mean square velocity of gas molecule is directly proportional to the square root of its absolute temperature.

#### Solution

Expression for r.m.s velocity:

a.   Let, P = pressure exerted by one mole of an ideal gas
V = volume of one mole of the gas
T = absolute temperature
b.   Pressure exerted by gas is given by,

P=1/3(Mc^2)/V

where M = mass of one mole (molecular weight) of the gas.

PV=1/3(Mc^2) .....................(1)

c.  But for one mole of an ideal gas, PV = RT

therefore RT=1/3 MC^2        [From Equation 1]

therefore c^2=(3RT)/M

therefore c=sqrt((3RT)/M)........................(2)

Equation (2) represents expression for r.m.s velocity of gas molecules.

d.   As R and M in equation (2) are constant,

therefore c prop sqrtT

therefore c_1/c_2=sqrt(T_1/T_2)

Is there an error in this question or solution?