Question

Hydrogen and oxygen gases have the same RMS speed. If hydrogen gas is at \(27^\circ\mathrm{C}\), find the temperature of oxygen gas.

• A. \(1200^\circ\mathrm{C}\)
• B. \(2400^\circ\mathrm{C}\)
• C. \(3600^\circ\mathrm{C}\)
• D. \(4527^\circ\mathrm{C}\)

Hint:

If two gases have the same RMS speed, temperature is directly proportional to molar mass \((T \propto M)\).

Answer 1

The Correct Option is D

Step 1: Write the formula for RMS speed.
The RMS speed of a gas is given by: \[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}}. \] For two gases having the same RMS speed: \[ \frac{T_1}{M_1} = \frac{T_2}{M_2}. \]
Step 2: Convert temperature into Kelvin.
Given temperature of hydrogen gas: \[ 27^\circ\mathrm{C} = 300\,\mathrm{K}. \]
Step 3: Substitute molar masses.
Molar mass of hydrogen gas \( (\mathrm{H_2}) = 2 \),
Molar mass of oxygen gas \( (\mathrm{O_2}) = 32 \).
\[ T_2 = T_1 \times \frac{M_2}{M_1} = 300 \times \frac{32}{2} = 300 \times 16 = 4800\,\mathrm{K}. \]
Step 4: Convert temperature back to Celsius.
\[ 4800\,\mathrm{K} = 4800 - 273 = 4527^\circ\mathrm{C}. \]
Step 5: Final conclusion.
The temperature of oxygen gas is: \[ \boxed{4527^\circ\mathrm{C}}. \]