Question

By what factor, the mean velocity is to be multiplied to get the RMS velocity?
(A) $\dfrac{{1.224}}{{1.128}}$
(B) $\dfrac{{1.128}}{{1.224}}$
(C) $\dfrac{1}{{1.128}}$
(D) $\dfrac{1}{{1.224}}$

Answer 1

Velocity is defined as the rate of change of position of an object with respect to a frame of reference and is a function of time. Its SI unit is meter per second while RMS velocity depends on the molecular mass and temperature of a molecule.
Formula Used: ${V_{rms}} = \sqrt {\dfrac{{3RT}}{M}}$
Where R is a constant, T is the temperature and M is the molecular mass.

Complete step by step solution:
The root mean square (RMS or rms) is defined as the root of the mean square. The RMS may be additionally referred to as the quadratic mean and is a particular case of the generalized mean with exponent. The mean velocity is the average of all the velocities of a particle.
The formula for root mean square velocity is given as follows,
${V_{rms}} = \sqrt {\dfrac{{3RT}}{M}}$
And the formula for the mean velocity is given as follows,
${V_{avg}} = \sqrt {\dfrac{{8RT}}{{\pi M}}}$
Here, ${V_{rms}}$ is the root mean square velocity and ${V_{avg}}$ is the mean velocity. T is the temperature measured in kelvin. M is the molar mass of the gas. The gas constant, also referred to as the molar gas constant, universal gas constant, or the perfect gas constant, is denoted by the symbol R. It's like the Boltzmann constant, but expressed in units of energy per temperature increment per mole, that is the pressure–volume product, instead of energy per temperature increment per particle.
By taking the ratio of both the formulas, we get the factor which is to be multiplied to mean velocity in order to get the RMS velocity.
Hence the mean velocity is to be multiplied with $\dfrac{{1.224}}{{1.128}}$ to get the RMS velocity.
Thus, the correct answer is option A.

Note:
There is also most probable velocity of the gas. This velocity is the velocity of the most of the gas molecules. The formula for most probable velocity is given as follows,
${V_{mp}} = \sqrt {\dfrac{{2RT}}{M}}$ .