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A 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.
1652.20 m/s
1847.25 m/s
2175.34 m/s
1254.20 m/s
Answers (1)
<p>Let <img alt="n_i" src="https://latex.codecogs.com/gif.latex?n_i" /> represent the number of particles with speed <img alt="v_i" src="https://latex.codecogs.com/gif.latex?v_i" />, where i denotes the speed category. The total number of particles, N, is 1100.<br />
The average speed ?v can be calculated using the formula:</p>
<p><img alt="\bar{v}=\frac{1}{N}\sum_i n_iv_i" src="https://latex.codecogs.com/gif.latex?%5Cbar%7Bv%7D%3D%5Cfrac%7B1%7D%7BN%7D%5Csum_i%20n_iv_i" /></p>
<p>Substituting the given values, we have:<br />
<img alt="\bar{v} =\frac{ 1}{1100}(1000.100 + 2000.200 + 4000.300 + 3000.400 + 1000 .500)" src="https://latex.codecogs.com/gif.latex?%5Cbar%7Bv%7D%20%3D%5Cfrac%7B%201%7D%7B1100%7D%281000.100%20&plus;%202000.200%20&plus;%204000.300%20&plus;%203000.400%20&plus;%201000%20.500%29" /></p>
<p> <img alt="=\frac{ 1}{1100}2200000" src="https://latex.codecogs.com/gif.latex?%3D%5Cfrac%7B%201%7D%7B1100%7D2200000" /></p>
<p> <img alt="=22000m/s" src="https://latex.codecogs.com/gif.latex?%3D22000m/s" /><br />
The root mean square (rms) speed <img alt="v_{rms }" src="https://latex.codecogs.com/gif.latex?v_%7Brms%20%7D" /> can be computed using the formula:</p>
<p><img alt="v_{rms } = vrms = \sqrt {\frac{1}{N}\sum _in_iv_i^2}" src="https://latex.codecogs.com/gif.latex?v_%7Brms%20%7D%20%3D%20vrms%20%3D%20%5Csqrt%20%7B%5Cfrac%7B1%7D%7BN%7D%5Csum%20_in_iv_i%5E2%7D" /></p>
<p>Substituting the given values, we get:<br />
<img alt="v_{rms } =\sqrt{\frac{1}{1100}(1000 . 1002 + 2000 . 2002 + 4000 .3002 + 3000 . 4002 + 1000 . 5002)}" src="https://latex.codecogs.com/gif.latex?v_%7Brms%20%7D%20%3D%5Csqrt%7B%5Cfrac%7B1%7D%7B1100%7D%281000%20.%201002%20&plus;%202000%20.%202002%20&plus;%204000%20.3002%20&plus;%203000%20.%204002%20&plus;%201000%20.%205002%29%7D" /></p>
<p> <img alt="=\sqrt{\frac{1}{1100} .5206000000}" src="https://latex.codecogs.com/gif.latex?%3D%5Csqrt%7B%5Cfrac%7B1%7D%7B1100%7D%20.5206000000%7D" /><br />
<img alt="\approx \sqrt{4732727.27}" src="https://latex.codecogs.com/gif.latex?%5Capprox%20%5Csqrt%7B4732727.27%7D" /><br />
<img alt="\approx2175.34 m/s." src="https://latex.codecogs.com/gif.latex?%5Capprox2175.34%20m/s." /><br />
Hence, the average speed of the gas system is approximately 2000 m/s, and the root mean square speed is approximately 2175.34 m/s.<br />
Therefore, the correct option is 3.</p>
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