Question

What will be the ratio between the root mean square speed of ${\mathrm{H}}_2$ at $50\mathrm{K}$ and that of ${\mathrm{O}}_2$ at $800\mathrm{K}$?

• A. $1$
• B. $2$
• C. $4$
• D. $1/4$

Answer 1

The correct option is A

$1$

Explanation for the correct option:

(A) $1$

Step 1: Apply the formula for the root mean square velocity for ${\mathrm{H}}_2$ at $50\mathrm{K}$

$$\begin{gathered} v_{rms}=\sqrt{\frac{3RT}{M}} \\ {v}_1=\sqrt{\frac{3R\times 50}{2}} \end{gathered}$$

Step 2: Apply the formula for the root mean square velocity for ${\mathrm{O}}_2$ at $800\mathrm{K}$

$v_2=\sqrt{\frac{3R\times 800}{32}}$

Step 3: Find the ratio of the speeds

$$\begin{gathered} \frac{v_1}{v_2}=\frac{\sqrt{{\displaystyle \frac{3R\times 50}{2}}}}{\sqrt{\frac{3R\times 800}{32}}} \\ \frac{v_1}{v_2}=\sqrt{\frac{50\times 32}{2\times 800}} \\ \frac{v_1}{v_2}=\sqrt{\frac{1600}{1600}}=1 \end{gathered}$$

Explanation for the incorrect options:

Since, the root mean square speed of ${\mathrm{H}}_2$ at $50\mathrm{K}$ and that of ${\mathrm{O}}_2$ at $800\mathrm{K}$ is $1$. Therefore, options (B), (C) and (D) stand incorrect.

Hence, option (A) is correct. The ratio between the root mean square speed of ${\mathrm{H}}_2$ at $50\mathrm{K}$ and that of ${\mathrm{O}}_2$ at $800\mathrm{K}$ is $1$.