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The average speed of molecules in an ideal gas is$$ \overline{v} = \frac{4}{\sqrt{\pi}} \left (\frac{M}{2RT} \right)^{\frac{3}{2}} \int_0^\infty v^3 e^{\frac{-Mv^2}{(2RT)}}\ dv $$where $ M $ is the molecular weight of the gas, $ R $ is the gas constant, $ T $ is the gas temperature, and $ v $ is the molecular speed. Show that$$ \overline{v} = \sqrt{\frac{8RT}{\pi M}} $$

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 8

Improper Integrals

Integration Techniques

Campbell University

University of Michigan - Ann Arbor

Boston College

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

07:27

The average speed of molec…

0:00

12:20

06:37

06:43

06:50

The average speed, $\bar{v…

03:25

02:15

The root mean square speed…

01:40

The probability that a mol…

07:57

According to Maxwell'…

02:05

For an ideal gas of $N$ mo…

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