**The kinetic theory of gases states that the kinetic energy of a gas is directly proportional…**

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## Question “The kinetic theory of gases states that the kinetic energy of a gas is directly proportional…”

The kinetic theory of gases states that the kinetic energy of a

gas is directly proportional to the temperature of the gas. A

relationship between the microscopic properties of the gas

molecules and the macroscopic properties of the gas can be derived

using the following assumptions: The gas is composed of pointlike

particles separated by comparatively large distances. The gas

molecules are in continual random motion with collisions being

perfectly elastic. The gas molecules exert no long-range forces on

each other. One of the most important microscopic properties of gas

molecules is velocity. There are several different ways to describe

statistically the average velocity of a molecule in a gas. The most

obvious measure is the average velocity Vavg. However, since the

molecules in a gas are moving in random directions, the average

velocity is approximately zero. Another measure of velocity is

(V^2)avg, the average squared velocity. Since the square of

velocity is always positive, this measure does not average to zero

over the entire gas. A third measure is the root-mean-square (rms)

speed,Vrms , equal to the square root of (V^2)avg. The rms speed is

a good approximation of the the typical speed of the molecules in a

gas. This histogram (Figure 1) shows a theoretical distribution of

speeds of molecules in a sample of nitrogen () gas. In this

problem, you’ll use the histogram to compute properties of the

gas.

A

B

Because the kinetic

energy of a single molecule is related to its velocity squared, the

best measure of the kinetic energy of the entire gas is obtained by

computing the mean squared velocity,, or its square root. The quantityis more common thanbecause it has the dimensions of velocity instead of

the less-familiar velocity-squared.

## Answer

This problem can be solved using the concepts of average and root mean square speed.

Find the average velocity for each point from the graph.

Next, determine the rms velocity of the nitrogen gas molecule by relating its kinetic energy to its velocity squared.

The root mean square speed expression can be used to calculate the temperature of the sample gases in the histogram.

The average is a middle value for some data.

Kinetic theory describes a gas as a large amount of submicroscopic particles.

The molecules have the same mass and are random in motion.

Root mean square speed is the measure of the speed of a particle in a gas.

Root mean square speed is the square root of average velocity divided by molecules in a gas.

For root mean square speed, the expression is:

refers to the gas constant,

refers to the temperature, HTMLmedia_tag_3$ refers to the mass of the molecule and HTMLmedia_tag_4$ the root mean square.

(A)

The diagram shows:

The average speed for a molecule is

(B)

The velocity square of a single molecule determines the kinetic energy.

As you can see,

(C)

For root mean square speed, the expression is:

Substitute

to

;

to

; and

to

.

Modify the expression above for

.

To convert Kelvin temperature to degrees, subtract 273.15 Celsius

Ans Part A

is the average speed of a molecule.

Part B

is the average speed of nitrogen gas molecules.

Part C

The histogram’s temperature is

.

## Conclusion

Above is the solution for “**The kinetic theory of gases states that the kinetic energy of a gas is directly proportional…**“. We hope that you find a good answer and gain the knowledge about this topic of **science**.