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  • A gas system contains 1100 particles, each with a specific speed distribution: • 1000 particles with a speed of 100 m/s each, • 2000 particles with a speed of 200 m/s each,<

A gas system contains 1100 particles, each with a specific 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

158.10 m/s
 


Option: 2

158.30 m/s
 


Option: 3

2175.34 m/s
 


Option: 4

2115.30 m/s


Answers (1)

best_answer

Let n_i represent the number of particles with speed v_i, where i denotes the speed category. The total number of particles, N, is 1100.
The average speed  ?v can be computed 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 vrms 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 obtain:

\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}Thus, 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.

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Gunjita

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