Question

Question: (II)If the pressure in a gas is tripled while its volume is held constant, by what factor does\({v_{{\rm{rms}}}}\)change?

Short answer

The rms speed changes by a factor of \(\sqrt 3 \).

Step-by-step solution

Understanding the ideal gas equation

Ideal gas law states that at constant volume, the pressure is directly proportional to the number of molecules and temperature of the gas.

The ideal gas equation is written as follows:

\(PV = nkT\)

Here, P is the pressure, V is the volume, n is the number of molecules, k is the Boltzmann constant, and T is the temperature.

Evaluation of the change in root mean square speed of the gas

At constant volume, when the pressure gets triples, the temperature will also be tripled.

\(T' = 3T\)

The rms speed of a gas molecule is directly proportional to the square root of the temperature.

\({v_{{\rm{rms}}}} \propto \sqrt T \)

Therefore,

\(\begin{aligned}{c}\frac{{{{v'}_{{\rm{rms}}}}}}{{{v_{{\rm{rms}}}}}} &= \sqrt {\frac{{T'}}{T}} \\\frac{{{{v'}_{{\rm{rms}}}}}}{{{v_{{\rm{rms}}}}}} &= \sqrt {\frac{{3T}}{T}} \\{{V'}_{{\rm{rms}}}} &= \sqrt 3 \times {v_{{\rm{rms}}}}\end{aligned}\)

Thus, the rms speed changes by a factor of \(\sqrt 3 \).