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  • Consider a gas system consisting of 1100 particles. The speed distribution of these particles is as follows:              

Consider a gas system consisting of 1100 particles. The speed distribution of these particles is as follows:

                                                    1000 particles each with speed 100 m/s,
                                                    2000 particles each with speed 200 m/s,
                                                    4000 particles each with speed 300 m/s,
                                                    3000 particles each with speed 400 m/s,
                                                    1000 particles each with speed 500 m/s.

Calculate the average speed and the root mean square (rms) speed of this gas system.

Option: 1

1812.45m/s


Option: 2

15.30 m/s


Option: 3

81.20 m/s


Option: 4

56.60 m/s


Answers (1)

best_answer

The average speed  ?v of a gas system can be calculated using the formula:

                                              \bar{v}=\frac{1}{N} \sum_i n_i v_i,

where N is the total number of particles, ni is the number of particles with speed v_i, and v_i is the speed of those particles.

In this case, we have N = 1100 particles. Substituting the given values:

\bar{v}=\frac{1}{1100}(1000 \times 100+2000 \times 200+4000 \times 300+3000 \times 400+1000 \times 500)

Calculating this gives:

\bar{v}=\frac{1}{1100} \times 2500000=2272.73 \mathrm{~m} / \mathrm{s}

The root mean square (rms) speed v_{rms} of the gas system is given by the formula:

v_{\mathrm{rms}}=\sqrt{\frac{1}{N} \sum_i n_i v_i^2} .

Substituting the given values:

v_{\mathrm{rms}}=\sqrt{\frac{1}{1100}\left(1000 \times 100^2+2000 \times 200^2+4000 \times 300^2+3000 \times 400^2+1000 \times 500^2\right)} .

Calculating this gives:

v_{\mathrm{rms}}=\sqrt{\frac{1}{1100} \times 3607000000}=\sqrt{3288181.82} \approx 1812.45 \mathrm{~m} / \mathrm{s}

Therefore, the average speed of the gas system is approximately 2272.73 m/s
and the root mean square speed is approximately 1812.45 m/s.

Therefore, the correct option is 1.

 

Posted by

shivangi.bhatnagar

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