Question

Molecular velocities of two gases at the same temperature are \[{U_1}\] and \[{U_2}\]and their molecular masses \[{m_1}\] and \[{m_2}\] respectively. Which of the following expressions is correct?
A. \[\dfrac{{{m_1}}}{{{U_1}^2}} = \dfrac{{{m_2}}}{{{U_2}^2}}\]
B. \[{m_1}{U_1} = {m_2}{U_2}\]
C. \[\dfrac{{{m_1}}}{{{U_1}}} = \dfrac{{{m_2}}}{{{U_2}}}\]
D. \[{m_1}{U_1}^2 = {m_2}{U_2}^2\]

Answer 1

Hint: The distance covered by a molecule of gas in one unit of time is called molecular velocity. The molecular velocities of different molecules are different because of differences in molecular masses of them.

Complete Step by Step Solution:
The formula used for calculation of molecular velocity is \[{u_{rms}} = \sqrt {\dfrac{{3RT}}{M}} \] , where, R stands for gas constant, T stands for temperature, M stands for molar mass of the gas.

 Here, two gases are given.
For gas 1,
Molecular mass = \[{m_1}\]
Molecular velocity = \[{U_1}\]

For gas 2,
Molecular mass = \[{m_2}\]
Molecular velocity = \[{U_2}\]

For 1st gas,
\[{u_1} = \sqrt {\dfrac{{3RT}}{{{m_1}}}} \]
\[T = \dfrac{{{u_1}^2{m_1}}}{{3RT}}\] …… (1)
For 2nd gas,
\[{u_2} = \sqrt {\dfrac{{3RT}}{{{m_2}}}} \]
\[T = \dfrac{{{u_2}^2{m_2}}}{{3RT}}\] …… (2)

From (1) and (2),
\[\dfrac{{{u_1}^2{m_1}}}{{3RT}} = \dfrac{{{u_2}^2{m_2}}}{{3RT}}\]
\[{u_1}^2{m_1} = {u_2}^2{m_2}\]
Therefore, option D is right.

Additional Information: Let’s understand how molecular mass is responsible for rms velocity. The rms velocity can be find out by the following formula, \[{v_{rms}} = \sqrt {\dfrac{{3RT}}{M}} \] .As R is gas constant and if given that all gases are at same temperature, molar masses of each gas decide its rms speed. So, \[{v_{rms}} = \sqrt {\dfrac{1}{M}} \], that means, rms velocity is indirectly proportional to molar mass of gases. So, the gas that possesses the highest molar mass has the lowest rms velocity.

Note: It is to be noted that average velocity is different from rms velocity. Average velocity defines the arithmetic mean calculation of velocities of different gaseous molecules at a particular temperature. The average velocity can be found by the formula \[{v_{av}} = \sqrt {\dfrac{{8RT}}{{\pi M}}} \] . The rms velocity is always greater than average velocity.