At 300 K, the rms speed of oxygen molecules is $\sqrt{\frac{\alpha+5}{\alpha}}$ times to that of its average speed in the gas. Then, the value of $\alpha$ will be (used $\pi=\frac{22}{7}$)

JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Heat & Thermodynamics

At 300 K, the rms speed of oxygen molecules is

\frac{α + 5}{α}

times to that of its average speed in the gas. Then, the value of

α

will be (used

π = \frac{22}{7}

)

(A)32
(B)28✓ Correct
(C)24
(D)27

Answer: (B)

Solution

The rms speed is

v_{r m s} = \frac{3 R T}{M}

and the average speed is

v_{a v g} = \frac{8 R T}{π M}

, so the square of their ratio is

\frac{v _{r m s}^{2}}{v _{a v g}^{2}} = \frac{3 π}{8}

. Setting this equal to

\frac{α + 5}{α}

and using

π = \frac{22}{7}

gives

\frac{α + 5}{α} = \frac{3 \times 22}{8 \times 7} = \frac{33}{28}

. Cross-multiplying,

28 (α + 5) = 33 α

, so

5 α = 140

and

α = 28

JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Single correct option

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