$\"\"$

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The average speed of molecules in an ideal gas is $\"\"$ .

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Where $\"\"$ is the molecular weight of the gas, $\"\"$ is the gas constant, $\"\"$ is the gas temperature and $\"\"$ is the molecular speed.

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$\"\"$

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Let $\"\"$ .

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$\"\"$ .

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$\"\"$

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Consider $\"\"$ .

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$\"\"$ .

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Solve the integral by using parts of integration method.

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Formula for integration by parts : $\"\"$ .

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$\"\"$ and $\"\"$ .

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Consider $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Consider $\"\"$ .

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Apply integral on each side.

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$\"\"$

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$\"\"$ .

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Substitute the corresponding values in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$ .