Question

At which temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at 47ºC?

• A. 80 K
• B. -73 K
• C. 20 K
• D. 4 K

Answer 1

Answer: (C)

20 K

Answer 2

Using the Formula for Root Mean Square (r.m.s.) Velocity: The r.m.s. velocity $v_{rms}$ for a gas is given by:

$v_{rms} = \sqrt{\frac{3RT}{M}}$

where $R$ is the gas constant, $T$ is the temperature, and $M$ is the molar mass of the gas.

Set up the Equation for Hydrogen and Oxygen: To find the temperature at which the r.m.s. velocity of hydrogen equals that of oxygen at 47°C, we set:

$\sqrt{\frac{3RT_{H_2}}{M_{H_2}}} = \sqrt{\frac{3RT_{O_2}}{M_{O_2}}}$

Isolate $T_{H_2}$: Square both sides to remove the square root:

$\frac{3RT_{H_2}}{M_{H_2}} = \frac{3RT_{O_2}}{M_{O_2}}$

Simplify by canceling $3R$ on both sides:

$T_{H_2} = T_{O_2} \times \frac{M_{H_2}}{M_{O_2}}$

Substitute Values for Molar Mass and Temperature: Given $T_{O_2} = 47°C = 320 \, K$,

$T_{H_2} = 320 \times \frac{2}{32} = 20 \, K$