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  • Consider a sample of gas containing 800 particles. The speed distribution of the particles is as follows: - 400 particles with speed <img alt="150 \mathrm{~m} / \mathrm{s}" src="https://e

Consider a sample of gas containing 800 particles. The speed distribution of the particles is as follows:

- 400 particles with speed 150 \mathrm{~m} / \mathrm{s} - 200 particles with speed 200 \mathrm{~m} / \mathrm{s} - 100 particles with speed 250 \mathrm{~m} / \mathrm{s} -50 particles with speed 300 \mathrm{~m} / \mathrm{s} -50 particles with speed 350 \mathrm{~m} / \mathrm{s} -100 particles with speed 400 \mathrm{~m} / \mathrm{s}

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

Option: 1

300 \mathrm{~m} / \mathrm{s}


Option: 2

250 \mathrm{~m} / \mathrm{s}


Option: 3

450 \mathrm{~m} / \mathrm{s}


Option: 4

200 \mathrm{~m} / \mathrm{s}


Answers (1)

best_answer

Step 1: Calculate the total number of particles (\mathrm{N})=800

Step 2: Calculate the average speed \left(v_{\text {avg }}\right)

n_{1}=400 (number of particles with speed \left.150 \mathrm{~m} / \mathrm{s}\right);v_{1}=150 \mathrm{~m} / \mathrm{s} n_2=200 (number of particles with speed $200 \mathrm{~m} / \mathrm{s}); v_{2}=200 \mathrm{~m} / \mathrm{s}; n_{3}=100 (number of particles with speed 250 \mathrm{~m} / \mathrm{s} ) ; v_{3}=250 \mathrm{~m} / \mathrm{s} ; n_{4}=50 (number of particles with speed 300 \mathrm{~m} / \mathrm{s} ) ; v_{4}=300 \mathrm{~m} / \mathrm{s} ; n_{5}=50 (number of particles with speed 350 \mathrm{~m} / \mathrm{s}); v_{5}=350 \mathrm{~m} / \mathrm{s} ; n_{6}=100 (number of particles with speed 400 \mathrm{~m} / \mathrm{s} ); v_{6}=400 \mathrm{~m} / \mathrm{s}

{Calculate the numerator}

Numerator =(400 \times 150)+(200 \times 200)+(100 \times 250)+(50 \times 300)+$ $(50 \times 350)+(100 \times 400)=60,000+40,000+25,000+15,000+17,500+$ $40,000=197,500

v_{\text {avg }}=$ Numerator $/ \mathrm{N}=197,500 / 800 \approx 246.88 \mathrm{~m} / \mathrm{s}

Step 3: Calculate the mean square speed \left(v_{\text {avg }}^{2}\right)

{Calculate the numerator}

Numerator \left(v_{\text {avg }}^{2}\right)=\left(400 \times 150^{2}\right)+\left(200 \times 200^{2}\right)+\left(100 \times 250^{2}\right)+\left(50 \times 300^{2}\right)$ $+\left(50 \times 350^{2}\right)+\left(100 \times 400^{2}\right)=9,000,000+8,000,000+6,250,000+4,500,000$ $+6,125,000+16,000,000=49,875,000

v_{\text {avg }}^{2}=$ Numerator $/ \mathrm{N}=49,875,000 / 800 \approx 62,343.75 \mathrm{~m}^{2} / \mathrm{s}^{2}

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

\text { rms speed }=\sqrt{v_{a v g}^{2}}=\sqrt{62,343.75}=249.69 \mathrm{~m} / \mathrm{s}

Answer:

The average speed of the gas particles is approximately 246.88 \mathrm{~m} / \mathrm{s}, and the root mean square (rms) speed is approximately 250 \mathrm{~m} / \mathrm{s}.

Posted by

shivangi.bhatnagar

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