Question

Find the ratio of the root mean square speed of oxygen gas molecules to that of hydrogen gas molecules, if the temperature of both gases is the same.

• A. \(\frac{1}{4}\)
• B. \(\frac{1}{16}\)
• C. \(\frac{1}{32}\)
• D. \(\frac{1}{8}\)

Answer 1

The Correct Option is A

\(V_{rms} = \sqrt{(\frac{3RT}{M})}\)
\(V_{rms}\propto \frac{1}{\sqrt{m}}\)
 \(\frac{(V_{rms})_{O_2}}{(V_{rms})_{H_2}} = \sqrt\frac{2}{32}= \frac{1}{4}\)

So, the correct option is (A): \(\frac 14\)

Answer 2

The root mean square (rms) speed of a gas molecule is given by:
\(V_{rms} = \sqrt{(\frac{3RT}{M})}\)
where k is the Boltzmann constant, T is the temperature, and m is the mass of the gas molecule.
Since the temperature of both oxygen and hydrogen gases is the same, we can write:\(\frac{V_{rms}O_2}{v_{rms}H_2} = \sqrt{\frac{mH_2}{mO_2}}\)
The molecular weight of hydrogen (H₂) is 2 grams per mole, while that of oxygen (O₂) is 32 grams per mole.
Substituting these values, we get:
\(\frac{V_{rms}O_2}{v_{rms}H_2} = \sqrt{\frac{32}{2}}=\sqrt{16}=4\)
Therefore, the ratio of the rms speed of oxygen gas molecules to that of hydrogen gas molecules is 4:1.
Hence, the answer is (A) \(\frac{1}{4}\).
Answer. A