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A $$25 \times 10^{-3}$$ 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?
Collision rate formula: $$Z_1 = \sqrt{2} \pi d^2 \bar{v} n$$
$$n = \frac{6.023 \times 10^{23}}{25 \times 10^{-3}} = 2.41 \times 10^{25}\text{ molecules/m}^3$$
$$Z_1 = \sqrt{2} \times \pi \times (0.3 \times 10^{-9})^2 \times 200 \times 2.41 \times 10^{25}$$
$$Z_1 \approx 1.93 \times 10^9\text{ s}^{-1}$$ $$\approx 0.2 \times 10^\text{10}\text{ s}^{-1}$$
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