Question

The temperature at which the rms speed of hydrogen molecules is equal to that of oxygen molecules at \(47^\circ C\) is

• A. \(20\,K\)
• B. \(80\,K\)
• C. \(73\,K\)
• D. \(3\,K\)

Hint:

For equal rms speeds: \[ \frac{T}{M}=\text{constant} \] Lighter gases require lower temperature to have same rms speed.

Answer 1

The Correct Option is A

Step 1: Recall the rms speed formula.
The root mean square speed of gas molecules is \[ v_{rms} = \sqrt{\frac{3RT}{M}} \] It depends on temperature $T$ and molar mass $M$.

Step 2: Set the two speeds equal.
We want hydrogen at temperature $T$ to have the same rms speed as oxygen at its temperature. Equal speed means \[ \frac{T_H}{M_H} = \frac{T_O}{M_O} \]
Step 3: Convert the oxygen temperature.
\[ T_O = 47^\circ\text{C} = 47 + 273 = 320\,\text{K} \]
Step 4: Put in the molar masses.
Hydrogen $M_H = 2$ and oxygen $M_O = 32$. \[ \frac{T}{2} = \frac{320}{32} \]
Step 5: Simplify the right side.
\[ \frac{T}{2} = 10 \]
Step 6: Solve for the temperature.
\[ T = 20\,\text{K} \] \[ \boxed{20\,\text{K}} \]