Question

The r.m.s. velocity of hydrogen molecules at temperature \( T \) is seven times the r.m.s. velocity of nitrogen molecules at 300 K. This temperature \( T \) is (Molecular weights of hydrogen and nitrogen are 2 and 28 respectively):

• A. 1050 K
• B. 1700 K
• C. 1350 K
• D. 2100 K

Hint:

The r.m.s. velocity is inversely proportional to the square root of the molecular mass. Use this relationship to compare r.m.s. velocities of different gases.

Answer 1

The Correct Option is A

Step 1: Using the formula for r.m.s velocity.
The r.m.s. velocity \( v_{\text{rms}} \) of a gas is given by: \[ v_{\text{rms}} = \sqrt{\frac{3kT}{m}} \] where \( k \) is Boltzmann's constant, \( T \) is temperature, and \( m \) is the molecular mass. For hydrogen and nitrogen, we know the ratio of r.m.s velocities: \[ \frac{v_{\text{rms}}(\text{H}_2)}{v_{\text{rms}}(\text{N}_2)} = \sqrt{\frac{m_{\text{N}_2}}{m_{\text{H}_2}}} \] Substituting the given values and solving for \( T \), we get: \[ T = 1050 \, \text{K} \] Step 2: Final Answer.
Thus, the temperature \( T \) is 1050 K.