Question

To have the same RMS value as that of hydrogen at 30K, what will be the temperature of the oxygen molecule?

• A. 60K
• B. 90K
• C. 180K
• D. 45K

Hint:

The RMS velocity is proportional to the square root of temperature and inversely proportional to the square root of the molecular mass. Hence, for heavier molecules, the temperature needs to be higher to achieve the same RMS velocity.

Answer 1

The Correct Option is A

The root mean square (RMS) speed of molecules is related to temperature by the formula: \[ v_{\text{rms}} = \sqrt{\frac{3 k T}{m}} \] Where: - \( v_{\text{rms}} \) is the root mean square velocity, - \( k \) is Boltzmann's constant, - \( T \) is the temperature, - \( m \) is the mass of the molecule. Since we want the RMS velocity of oxygen molecules to be the same as that of hydrogen at 30K, we can set up the equation: \[ \frac{v_{\text{rms, O2}}}{v_{\text{rms, H2}}} = \sqrt{\frac{T_{\text{O2}}}{T_{\text{H2}}}} \times \sqrt{\frac{m_{\text{H2}}}{m_{\text{O2}}}} \] Using the fact that the mass of oxygen is approximately 16 times the mass of hydrogen, we get: \[ 1 = \sqrt{\frac{T_{\text{O2}}}{30}} \times \sqrt{\frac{2}{32}} \] Solving for \( T_{\text{O2}} \), we find: \[ T_{\text{O2}} = 60K \] Thus, the temperature required for the oxygen molecule to have the same RMS speed as hydrogen at 30K is 60K.