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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.

A green vector v, representing a molecule colliding with a wall, is pointing at the surface of a wall at an angle. A second vector v primed starts at the point of impact and travels away from the wall at an angle. A dotted line perpendicular to the wall through the point of impact represents the component of the molecule’s momentum that is perpendicular to the wall. A red vector F is pointing into the wall from the point of impact, representing the force of the molecule hitting the wall.
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:

PV = 1 3 Nm v 2 ¯ , size 12{ ital "PV"= { {1} over {3} } ital "Nm" {overline {v rSup { size 8{2} } }} ,} {}

where P size 12{P} {} is the pressure (average force per unit area), V size 12{V} {} is the volume of gas in the container, N size 12{N} {} is the number of molecules in the container, m size 12{m} {} is the mass of a molecule, and v 2 ¯ size 12{ {overline {v rSup { size 8{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:

PV = NkT . size 12{ ital "PV"= ital "NkT"} {}

Equating the right-hand side of this equation with the right-hand side of PV = 1 3 Nm v 2 ¯ size 12{ ital "PV"= { {1} over {3} } ital "Nm" {overline {v rSup { size 8{2} } }} } {} gives

1 3 Nm v 2 ¯ = NkT . size 12{ { {1} over {3} } ital "Nm" {overline {v rSup { size 8{2} } }} = ital "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.

Diagram representing the pressures that a gas exerts on the walls of a box in a three-dimensional coordinate system with x, y, and z components.
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 size 12{x} {} -direction, its momentum changes from mv x size 12{– ital "mv" rSub { size 8{x} } } {} to + mv x size 12{+ ital "mv" rSub { size 8{x} } } {} . Thus, its change in momentum is Δ mv = + mv x mv x = 2 mv x size 12{Δ ital "mv""=+" ital "mv" rSub { size 8{x} } – left (– ital "mv" rSub { size 8{x} } right )=2 ital "mv" rSub { size 8{x} } } {} . The force exerted on the molecule is given by

F = Δ p Δ t = 2 mv x Δ t . size 12{F= { {Δp} over {Δt} } = { {2 ital "mv" rSub { size 8{x} } } over {Δt} } "." } {}

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 Δ t size 12{Dt} {} 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 2 l ) size 12{2l \) } {} at a speed of v x size 12{v rSub { size 8{x} } } {} . Thus Δ t = 2 l / v x size 12{Δt=2l/v rSub { size 8{x} } } {} , and the expression for the force becomes

F = 2 mv x 2 l / v x = mv x 2 l . size 12{F= { {2 ital "mv" rSub { size 8{x} } } over { {2l} slash {v rSub { size 8{x} } } } } = { { ital "mv" rSub { size 8{x} } rSup { size 8{2} } } over {l} } "." } {}

This force is due to one molecule. We multiply by the number of molecules N size 12{N} {} and use their average squared velocity to find the force

F = N m v x 2 ¯ l , size 12{F=N { {m {overline {v rSub { size 8{x} } rSup { size 8{2} } }} } over {l} } ,} {}

where the bar over a quantity means its average value. We would like to have the force in terms of the speed v size 12{v} {} , rather than the x size 12{x} {} -component of the velocity. We note that the total velocity squared is the sum of the squares of its components, so that

v 2 ¯ = v x 2 ¯ + v y 2 ¯ + v z 2 ¯ . size 12{ {overline {v rSup { size 8{2} } }} = {overline {v rSub { size 8{x} } rSup { size 8{2} } }} + {overline {v rSub { size 8{y} } rSup { size 8{2} } }} + {overline {v rSub { size 8{z} } rSup { size 8{2} } }} "." } {}

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

v x 2 ¯ = v y 2 ¯ = v z 2 ¯ . size 12{ {overline {v rSub { size 8{x} } rSup { size 8{2} } }} = {overline {v rSub { size 8{y} } rSup { size 8{2} } }} = {overline {v rSub { size 8{z} } rSup { size 8{2} } }} "." } {}

Thus,

v 2 ¯ = 3 v x 2 ¯ , size 12{ {overline {v rSup { size 8{2} } }} =3 {overline {v rSub { size 8{x} } rSup { size 8{2} } }} ,} {}

or

v x 2 ¯ = 1 3 v 2 ¯ . size 12{ {overline {v rSub { size 8{x} } rSup { size 8{2} } }} = { {1} over {3} } {overline {v rSup { size 8{2} } }} } {}

Substituting 1 3 v 2 ¯ size 12{ { {1} over {3} } {overline {v rSup { size 8{2} } }} } {} into the expression for F size 12{F} {} gives

F = N m v 2 ¯ 3 l . size 12{F=N { {m {overline {v rSup { size 8{2} } }} } over {3l} } "." } {}

The pressure is F / A , size 12{F/A,} {} so that we obtain

P = F A = N m v 2 ¯ 3 Al = 1 3 Nm v 2 ¯ V , size 12{P= { {F} over {A} } =N { {m {overline {v rSup { size 8{2} } }} } over {3 ital "Al"} } = { {1} over {3} } { { ital "Nm" {overline {v rSup { size 8{2} } }} } over {V} } ,} {}

where we used V = Al size 12{V= ital "Al"} {} for the volume. This gives the important result.

PV = 1 3 Nm v 2 ¯ size 12{ ital "PV"= { {1} over {3} } ital "Nm" {overline {v rSup { size 8{2} } }} } {}

This equation is another expression of the ideal gas law.

Questions & Answers

What is conductivity
Saud Reply
It is the ease with which electrical charges or heat can be transmitted through a material or a solution.
Cffrrcvccgg
how to find magnitude and direction
Arjune Reply
how to caclculate for speed
Arjune
derivation of ohms law
Kazeem Reply
derivation of resistance
Kazeem
R=v/I where R=resistor, v=voltage, I=current
Kazeem
magnitude
Arjune
A puck is moving on an air hockey table. Relative to an x, y coordinate system at time t 0 s, the x components of the puck’s ini￾tial velocity and acceleration are v0x 1.0 m/s and ax 2.0 m/s2 . The y components of the puck’s initial velocity and acceleration are v0y 2.0 m/s and ay 2.0
Arjune
Electric current is the flow of electrons
Kelly Reply
is there really flow of electrons exist?
babar
Yes It exists
Cffrrcvccgg
explain plz how electrons flow
babar
if electron flows from where first come and end the first one
babar
an electron will flow accross a conductor because or when it posseses kinectic energy
Cffrrcvccgg
electron can not flow jist trasmit electrical energy
ghulam
free electrons of conductor
ankita
electric means the flow heat current.
Serah Reply
electric means the flow of heat current in a circuit.
Serah
What is electric
Manasseh Reply
electric means?
ghulam
electric means the flow of heat current in a circuit.
Serah
a boy cycles continuously through a distance of 1.0km in 5minutes. calculate his average speed in ms-1(meter per second). how do I solve this
Jenny Reply
speed = distance/time be sure to convert the km to m and minutes to seconds check my utube video "mathwithmrv speed"
PhysicswithMrV
d=1.0km÷1000=0.001 t=5×60=300s s=d\t s=0.001/300=0.0000033m\s
Serah
A puck is moving on an air hockey table. Relative to an x, y coordinate system at time t 0 s, the x components of the puck’s ini￾tial velocity and acceleration are v0x 1.0 m/s and ax 2.0 m/s2 . The y components of the puck’s initial velocity and acceleration are v0y 2.0 m/s and ay 2.0
Arjune
why we cannot use DC instead of AC in a transformer
kusshaf Reply
becuse the d .c cannot travel for long distance trnsmission
ghulam
what is physics
Chiwetalu Reply
branch of science which deals with matter energy and their relationship between them
ghulam
Life science
the
what is heat and temperature
Kazeem Reply
how does sound affect temperature
Clement Reply
sound is directly proportional to the temperature.
juny
how to solve wave question
Wisdom Reply
I would like to know how I am not at all smart when it comes to math. please explain so I can understand. sincerly
Emma
Just know d relationship btw 1)wave length 2)frequency and velocity
Talhatu
First of all, you are smart and you will get it👍🏽... v = f × wavelength see my youtube channel: "mathwithmrv" if you want to know how to rearrange equations using the balance method
PhysicswithMrV
nice self promotion though xD
Beatrax
thanks dear
Chuks
hi pls help me with this question A ball is projected vertically upwards from the top of a tower 60m high with a velocity of 30ms1.what is the maximum height above the ground level?how long does it take to reach the ground level?
mahmoud
what is scalar quantities
babatunde
scaler quantity are quanties that have only direction and no magnitude
Natsu
ice Point
babatunde
please guys help, what is the difference between concave lens and convex lens
Vincent Reply
convex lens brings rays of light to a focus while concave diverges rays of light
Christian
for mmHg to kPa yes
Matthew
it depends on the size
Matthew Reply
please what is concave lens
Vincent
a lens which diverge the ray of light
rinzuala
concave diverges light
Matthew
thank you guys
Vincent
A diverging lens
Yusuf
What is isotope
Yusuf
each of two or more forms of the same element that contain equal numbers of protons but different numbers of neutrons in their nuclei, and hence differ in relative atomic mass but not in chemical properties; in particular, a radioactive form of an element. "some elements have only one stable isotope
Karthi
what is wire wound resistors?
Naveedkhan Reply
Practice Key Terms 1

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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