Question

At a given temperature, what is the root mean square velocity of a gas molecule of mass $m$ proportional to?
A. ${m^0}$
B. $m$
C. $\sqrt m $
D. $\dfrac{1}{{\sqrt m }}$

Answer 1

Hint:Root mean square velocity of individual gas molecules is given by ${v_{rms}} = \sqrt {\dfrac{{3RT}}{M}} $ , where T is the temperature of gas molecules in Kelvin, M is the molar mass of the gas and $R$ is the universal gas constant having a value of $8.314{\text{ J}}{{\text{K}}^{ - 1}}{\text{mo}}{{\text{l}}^{ - 1}}$ . To solve the above question, use this formula of root mean square velocity.



Formula used:
 RMS velocity of gas molecules is given by:
${v_{rms}} = \sqrt {\dfrac{{3RT}}{M}} $
Here, T is the temperature of gas molecules in Kelvin,
M is the molar mass of the gas and
$R$ is the universal gas constant having a value of $8.314{\text{ J}}{{\text{K}}^{ - 1}}{\text{mo}}{{\text{l}}^{ - 1}}$.


Complete answer:
RMS velocity of gas molecules is given by:
${v_{rms}} = \sqrt {\dfrac{{3RT}}{M}} $ … (1)
Now, we know that Molar mass of a gas molecule is given by the ratio of molecular mass, $m$ of the gas and number of moles, $n$ , that is,
$M = \dfrac{m}{n}$ … (2)
Substituting this in equation (1),
${v_{rms}} = \sqrt {\dfrac{{3RTn}}{m}} $
From the above relation, we can conclude that ${v_{rms}}$ is directly proportional to $\dfrac{1}{{\sqrt m }}$ .
Thus, the correct option is D.



Note: For individual gas molecules, root mean square velocity is calculated as ${v_{rms}} = \sqrt {\dfrac{{3RT}}{M}} $ . Root Mean Square velocity is directly proportional to the absolute temperature of the gas and inversely proportional to the molar mass.