Question

The r.m.s velocity of CO gas molecules at $27{}^\circ C$ is approximately 1000 m/s. For ${{N}_{2}}$ molecules at 600 K, the r.m.s velocity is approximately:
(A)- 2000 m/s
(B)- 1414 m/s
(C)- 1000 m/s
(D)- 1500 m/s

Answer 1

The atoms or molecules move at different speeds and in random directions. The root mean square (r.m.s) velocity is the average velocity of all the gas particles, and this can be calculated using the formula-
${{\mu }_{rms}}=\sqrt{\dfrac{3RT}{m}}$
where ${{\mu }_{rms}}$is the root mean square velocity in m/sec
R is the ideal gas constant = 8.3145 $(kg.{{m}^{2}}/{{\sec }^{2}})/K.mol$
T is the absolute temperature in Kelvin
m is the mass of a mole of the gas in kilograms

Complete answer:
-In the relation between the temperature and the kinetic energy, we can relate temperature to the velocity of gas molecules.
${{E}_{k}}=\dfrac{3}{2}RT...(i)$
We know that, ${{E}_{k}}=\dfrac{1}{2}m{{v}^{2}}...(ii)$
-Equating (i) and (ii), for the value of velocity, we will get
${{v}^{2}}=\dfrac{3RT}{m}$
-The root mean square velocity or ${{v}_{rms}}$is the square root of the average square velocity, i.e.
${{v}_{rms}}=\sqrt{\dfrac{3RT}{m}}$
-Here, according to the question,
${{v}_{rms}}$for CO = 1000 m/s
Molar mass of CO = 12 + 16 = 28 g/mol $=2.8\times {{10}^{-2}}kg/mol$
Molar mass of ${{N}_{2}}$ = $14\times 14=28$ g/mol $=2.8\times {{10}^{-2}}kg/mol$
${{(T)}_{CO}}$ = 27 + 273 = 300 K
${{(T)}_{{{N}_{2}}}}$ = 600 K
-Plugging in the values in the equation,
$\dfrac{{{\mu }_{rms,{{N}_{2}}}}}{{{\mu }_{rms,CO}}}=\sqrt{\dfrac{{{(T)}_{{{N}_{2}}}}{{M}_{CO}}}{{{(T)}_{CO}}{{M}_{{{N}_{2}}}}}}$
$\Rightarrow \dfrac{{{\mu }_{rms,{{N}_{2}}}}}{1000}=\sqrt{\dfrac{600\times 2.8\times {{10}^{-2}}}{300\times 2.8\times {{10}^{-2}}}}$
Hence, ${{\mu }_{rms,{{N}_{2}}}}=1414m/s$

So, the correct answer is option B.

Note:
The root-mean-square speed gives the root mean square speed, not velocity. This is because velocity is a vector quantity, and hence it has magnitude as well as direction. The root-mean-square speed gives only the magnitude or speed. The temperature must be converted to Kelvin and the molar mass must be converted to kg/mol unit.