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

The Correct Option is C

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 \( T_{O_2} = 47°C = 320 \, K \),

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