# 13.4 Kinetic theory: atomic and molecular explanation of pressure

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• Express the ideal gas law in terms of molecular mass and velocity.
• Define thermal energy.
• Calculate the kinetic energy of a gas molecule, given its temperature.
• Describe the relationship between the temperature of a gas and the kinetic energy of atoms and molecules.
• Describe the distribution of speeds of molecules in a gas.

We have developed macroscopic definitions of pressure and temperature. Pressure is the force divided by the area on which the force is exerted, and temperature is measured with a thermometer. We gain a better understanding of pressure and temperature from the kinetic theory of gases, which assumes that atoms and molecules are in continuous random motion. When a molecule collides with a rigid wall, the component of its momentum perpendicular to the wall is reversed. A force is thus exerted on the wall, creating pressure.

[link] shows an elastic collision of a gas molecule with the wall of a container, so that it exerts a force on the wall (by Newton’s third law). Because a huge number of molecules will collide with the wall in a short time, we observe an average force per unit area. These collisions are the source of pressure in a gas. As the number of molecules increases, the number of collisions and thus the pressure increase. Similarly, the gas pressure is higher if the average velocity of molecules is higher. The actual relationship is derived in the Things Great and Small feature below. The following relationship is found:

$\text{PV}=\frac{1}{3}\text{Nm}\overline{{v}^{2}},$

where $P$ is the pressure (average force per unit area), $V$ is the volume of gas in the container, $N$ is the number of molecules in the container, $m$ is the mass of a molecule, and $\overline{{v}^{2}}$ is the average of the molecular speed squared.

What can we learn from this atomic and molecular version of the ideal gas law? We can derive a relationship between temperature and the average translational kinetic energy of molecules in a gas. Recall the previous expression of the ideal gas law:

$\text{PV}=\text{NkT}.$

Equating the right-hand side of this equation with the right-hand side of $\text{PV}=\frac{1}{3}\text{Nm}\overline{{v}^{2}}$ gives

$\frac{1}{3}\text{Nm}\overline{{v}^{2}}=\text{NkT}.$

## Making connections: things great and small—atomic and molecular origin of pressure in a gas

[link] shows a box filled with a gas. We know from our previous discussions that putting more gas into the box produces greater pressure, and that increasing the temperature of the gas also produces a greater pressure. But why should increasing the temperature of the gas increase the pressure in the box? A look at the atomic and molecular scale gives us some answers, and an alternative expression for the ideal gas law.

The figure shows an expanded view of an elastic collision of a gas molecule with the wall of a container. Calculating the average force exerted by such molecules will lead us to the ideal gas law, and to the connection between temperature and molecular kinetic energy. We assume that a molecule is small compared with the separation of molecules in the gas, and that its interaction with other molecules can be ignored. We also assume the wall is rigid and that the molecule’s direction changes, but that its speed remains constant (and hence its kinetic energy and the magnitude of its momentum remain constant as well). This assumption is not always valid, but the same result is obtained with a more detailed description of the molecule’s exchange of energy and momentum with the wall. Gas in a box exerts an outward pressure on its walls. A molecule colliding with a rigid wall has the direction of its velocity and momentum in the x size 12{x} {} -direction reversed. This direction is perpendicular to the wall. The components of its velocity momentum in the y size 12{y} {} - and z size 12{z} {} -directions are not changed, which means there is no force parallel to the wall.

If the molecule’s velocity changes in the $x$ -direction, its momentum changes from $–{\text{mv}}_{x}$ to $+{\text{mv}}_{x}$ . Thus, its change in momentum is $\text{Δ}\text{mv}\phantom{\rule{0.20em}{0ex}}\text{= +}{\text{mv}}_{x}–\left(–{\text{mv}}_{x}\right)=2{\text{mv}}_{x}$ . The force exerted on the molecule is given by

$F=\frac{\text{Δ}p}{\text{Δ}t}=\frac{2{\text{mv}}_{x}}{\text{Δ}t}\text{.}$

There is no force between the wall and the molecule until the molecule hits the wall. During the short time of the collision, the force between the molecule and wall is relatively large. We are looking for an average force; we take $\text{Δ}t$ to be the average time between collisions of the molecule with this wall. It is the time it would take the molecule to go across the box and back (a distance $2l\right)$ at a speed of ${v}_{x}$ . Thus $\text{Δ}t=2l/{v}_{x}$ , and the expression for the force becomes

$F=\frac{2{\text{mv}}_{x}}{2l/{v}_{x}}=\frac{{\text{mv}}_{x}^{2}}{l}\text{.}$

This force is due to one molecule. We multiply by the number of molecules $N$ and use their average squared velocity to find the force

$F=N\frac{m\overline{{v}_{x}^{2}}}{l},$

where the bar over a quantity means its average value. We would like to have the force in terms of the speed $v$ , rather than the $x$ -component of the velocity. We note that the total velocity squared is the sum of the squares of its components, so that

$\overline{{v}^{2}}=\overline{{v}_{x}^{2}}+\overline{{v}_{y}^{2}}+\overline{{v}_{z}^{2}}\text{.}$

Because the velocities are random, their average components in all directions are the same:

$\overline{{v}_{x}^{2}}=\overline{{v}_{y}^{2}}=\overline{{v}_{z}^{2}}\text{.}$

Thus,

$\overline{{v}^{2}}=3\overline{{v}_{x}^{2}},$

or

$\overline{{v}_{x}^{2}}=\frac{1}{3}\overline{{v}^{2}}.$

Substituting $\frac{1}{3}\overline{{v}^{2}}$ into the expression for $F$ gives

$F=N\frac{m\overline{{v}^{2}}}{3l}\text{.}$

The pressure is $F/A,$ so that we obtain

$P=\frac{F}{A}=N\frac{m\overline{{v}^{2}}}{3\text{Al}}=\frac{1}{3}\frac{\text{Nm}\overline{{v}^{2}}}{V},$

where we used $V=\text{Al}$ for the volume. This gives the important result.

$\text{PV}=\frac{1}{3}\text{Nm}\overline{{v}^{2}}$

This equation is another expression of the ideal gas law.

#### Questions & Answers

the range of objects and phenomena studied in physics is
Bethel Reply
what is Linear motion
Hamza Reply
straight line motion is called linear motion
then what
Amera
linear motion is a motion in a line, be it in a straight line or in a non straight line. It is the rate of change of distance.
Saeedul
Hi
aliyu
your are wrong Saeedul
Richard
Linear motion is a one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension
Jason
what is the formula to calculate wavelength of the incident light
David Reply
if a spring is is stiffness of 950nm-1 what work will be done in extending the spring by 60mmp
Hassan Reply
State the forms of energy
Samzy Reply
machanical
Ridwan
Word : Mechanical wave Definition : The waves, which need a material medium for their propagation, e.g., Sound waves. \n\nOther Definition: The waves, which need a material medium for their propagation, are called mechanical waves. Mechanical waves are also called elastic waves. Sound waves, water waves are examples of mechanical waves.t Definition: wave consisting of periodic motion of matter; e.g. sound wave or water wave as opposed to electromagnetic wave.h
Clement Reply
correct
Akinpelu
what is mechanical wave
Akinpelu Reply
a wave which require material medium for its propagation
syed
The S.I unit for power is what?
Samuel Reply
watt
Okoli
Am I correct
Okoli
it can be in kilowatt, megawatt and so
Femi
yes
Femi
correct
Jaheim
kW
Akinpelu
OK that's right
Samuel
SI.unit of power is.watt=j/c.but kw.and Mw are bigger.umots
syed
What is physics
aish Reply
study of matter and its nature
Akinpelu
The word physics comes from a Greek word Physicos which means Nature.The Knowledge of Nature. It is branch of science which deals with the matter and energy and interaction between them.
Uniform
why in circular motion, a tangential acceleration can change the magnitude of the velocity but not its direction
Syafiqah Reply
reasonable
Femi
because it is balanced by the inward acceleration otherwise known as centripetal acceleration
MUSTAPHA
What is a wave
Mutuma Reply
Tramsmission of energy through a media
Mateo
is the disturbance that carry materials as propagation from one medium to another
Akinpelu
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Akinpelu
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Favour Reply
Difference between north seeking pole and south seeking pole
Stanley Reply
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Swami Reply
please formula for density is what
Akinpelu
mass/volume
Jaja
what is motion
Nelson Reply
In physics, motion is the change in position of an object with respect to its surroundings in a given interval of time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, time, and speed. ... Momentum is a quantity which is used for measuring the motion of
Karthi

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