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

The root mean square (r.m.s.) velocity \( v_{rms} \) of a gas is calculated using the formula:

\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

where \( R \) is the gas constant, \( T \) is the temperature, and \( M \) is the molar mass of the gas.

To determine the temperature at which the r.m.s. velocity of hydrogen gas is equivalent to that of oxygen gas at 47°C, the following equation is established:

\[ \sqrt{\frac{3RT_{H_2}}{M_{H_2}}} = \sqrt{\frac{3RT_{O_2}}{M_{O_2}}} \]

To isolate \( T_{H_2} \), both sides of the equation are squared:

\[ \frac{3RT_{H_2}}{M_{H_2}} = \frac{3RT_{O_2}}{M_{O_2}} \]

Simplifying by canceling the \( 3R \) term from both sides yields:

\[ T_{H_2} = T_{O_2} \times \frac{M_{H_2}}{M_{O_2}} \]

Substituting the given values, where \( T_{O_2} = 47°C = 320 \, K \), and the molar masses of hydrogen (\( M_{H_2} = 2 \)) and oxygen (\( M_{O_2} = 32 \)):

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