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  • Consider a container of gas with 1200 molecules. The speed distribution of the molecules is as follows: - 800 molecules with speed <img alt="200 \mathrm{~m} / \mathrm{s}" src="https://ent

Consider a container of gas with 1200 molecules. The speed distribution of the molecules is as follows:

- 800 molecules with speed 200 \mathrm{~m} / \mathrm{s} - 300 molecules with speed 300 \mathrm{~m} / \mathrm{s} 100 molecules with speed 400 \mathrm{~m} / \mathrm{s}

Find the average speed and the root mean square (rms) speed of the gas molecules.

Option: 1

\operatorname{Vav}=242 \mathrm{~m} / \mathrm{s}, \operatorname{Vrms}=250 \mathrm{~m} / \mathrm{s}


Option: 2

\operatorname{Vav}=250 \mathrm{~m} / \mathrm{s}$, Vrms $=242 \mathrm{~m} / \mathrm{s}


Option: 3

\operatorname{Vav}=270 \mathrm{~m} / \mathrm{s}$, Vrms $=250 \mathrm{~m} / \mathrm{s}


Option: 4

\operatorname{Vav}=242 \mathrm{~m} / \mathrm{s}, \operatorname{Vrms}=240 \mathrm{~m} / \mathrm{s}


Answers (1)

best_answer

Step 1: Calculate the total number of molecules (\mathrm{N})=1200

Step 2: Calculate the average speed \left(v_{a v g}\right)

n_{1}=800 (number of molecules with speed $200 \mathrm{~m} / \mathrm{s}) ; v_{1}=200 \mathrm{~m} / \mathrm{s} ; n_{2}=300 ( number of molecules with speed 300 \mathrm{~m} / \mathrm{s}) ; v_{2}=300 \mathrm{~m} / \mathrm{s}; n_{3}=100 (number of molecules with speed 400 \mathrm{~m} / \mathrm{s} ) ; v_{3}=400 \mathrm{~m} / \mathrm{s}

Calculate the numerator

Numerator =(800 \times 200)+(300 \times 300)+(100 \times 400)=160,000+90,000$ $+40,000=290,000

v_{\text {avg }}=$ Numerator $/ \mathrm{N}=290,000 / 1200 \approx 241.67 \mathrm{~m} / \mathrm{s}

Step 3: Calculate the mean square speed \left(v_{a v g}^{2}\right)

Calculate the numerator

Numerator \left(v_{a v g}^{2}\right)=\left(800 \times 200^{2}\right)+\left(300 \times 300^{2}\right)+\left(100 \times 400^{2}\right)=32,000,000$ $+27,000,000+16,000,000=75,000,000

v_{a v g}^{2}=$ Numerator $/ \mathrm{N}=75,000,000 / 1200 \approx 62,500 \mathrm{~m}^{2} / \mathrm{s}^{2}

Step 4: Calculate the root mean square (rms) speed

rms speed =\sqrt{v_{a v g}^{2}}=\sqrt{62,500} \approx 250 \mathrm{~m} / \mathrm{s}

Answer:

The average speed of the gas molecules is approximately 241.67 \mathrm{~m} / \mathrm{s}, and the root mean square (rms) speed is approximately 250 \mathrm{~m} / \mathrm{s}.

Posted by

Suraj Bhandari

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