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  • An ideal gas system consists of 1100 particles with the following speed distribution: • 1000 particles with a speed of 100 m/s each, • 2000 particles with a speed of 200

An ideal gas system consists of 1100 particles with the following speed distribution:

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

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

Option: 1

891.60 m/s


Option: 2

154.170 m/s


Option: 3

2175.34 m/s


Option: 4

1812.45 m/s


Answers (1)

Let’s denote the number of particles with speed vi as ni, where i represents the speed category. The total number of particles, N, is 1100.

The average speed  ?v can be calculated using the formula:

                                                \bar{v}=\frac{1}{N} \sum_i n_i v_i .
.
Substituting the given values, we have:

 

\begin{aligned} \bar{v} & =\frac{1}{1100}(1000 \cdot 100+2000 \cdot 200+4000 \cdot 300+3000 \cdot 400+1000 \cdot 500) \\ & =\frac{1}{1100} \cdot 2200000 \\ & \approx 2000 \mathrm{~m} / \mathrm{s} . \end{aligned}The root mean square (rms) speed v_{rms} can be calculated using the formula:

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

Substituting the given values, we get:

\begin{aligned} v_{\mathrm{rms}} & =\sqrt{\frac{1}{1100}\left(1000 \cdot 100^2+2000 \cdot 200^2+4000 \cdot 300^2+3000 \cdot 400^2+1000 \cdot 500^2\right)} \\ & =\sqrt{\frac{1}{1100} \cdot 5206000000} \\ & \approx \sqrt{4732727.27} \\ & \approx 2175.34 \mathrm{~m} / \mathrm{s} . \end{aligned}

Therefore, the average speed of the gas system is approximately 2000 m/s, and the root mean square speed is approximately 2175.34 m/s.

Therefore, the correct option is 3.

Posted by

Kshitij

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