A $25 \times 10^{-3} \mathrm{~m}^{3}$ volume cylinder is filled with 1 mol of $\mathrm{O}_{2}$ gas at room temperature ( 300 K ). The molecular diameter of $\mathrm{O}_{2}$, and its root mean square speed, are found to be 0.3 nm and $200 \mathrm{~m} / \mathrm{s}$, respectively. What is the average collision rate (per second) for an $\mathrm{O}_{2}$ molecule?

JEE Main · 2019 · 10 Apr 2019 · Shift 1 · Heat & Thermodynamics

25 \times 1 0^{- 3} m^{3}

volume cylinder is filled with 1 mol of

O_{2}

gas at room temperature ( 300 K ). The molecular diameter of

O_{2}

, and its root mean square speed, are found to be 0.3 nm and

200 m / s

, respectively. What is the average collision rate (per second) for an

O_{2}

molecule?

Answer: (A)

The collision rate is

ν = \frac{v _{a v}}{λ}

with mean free path

λ = \frac{V}{2 π σ ^{2} N _{A}}

, where

σ = 0.6 \times 1 0^{- 9}

m. The average speed is

v_{a v} = \frac{8}{3 π} v_{r m s} \approx 0.92 \times 200 = 184

m/s. Substituting,

ν = \frac{184 \times 2 π ( 0.6 \times 1 0 ^{- 9} ) ^{2} \times 6.02 \times 1 0 ^{23}}{25 \times 1 0 ^{- 3}}

, which comes out on the order of

1 0^{10}

collisions per second.

JEE Main · 2019 · 10 Apr 2019 · Shift 1 · Single correct option