KINETIC THEORY

• Consider an ideal gas with molecules that are a point mass.

KINETIC THEORY

• Consider an ideal gas with molecules that are a point mass. The gas is confined in a cubic volume

KINETIC THEORY

• Consider an ideal gas with molecules that are a point mass. The gas is confined in a cubic volume and the density of molecules is rare so that there are very few molecular-molecular collisions

KINETIC THEORY

• Consider an ideal gas with molecules that are a point mass. The gas is confined in a cubic volume and the density of molecules is rare so that there are very few molecular-molecular collisions and there are no intermolecular forces.

KINETIC THEORY

• Consider an ideal gas with molecules that are a point mass. The gas is confined in a cubic volume and the density of molecules is rare so that there are very few molecular-molecular collisions and there are no intermolecular forces. When the molecules strike the inside of the container there are elastic collisions.

KINETIC THEORY

• The microscopic movement of molecules is used to describe macroscopic parameters.

KINETIC THEORY

• The microscopic movement of molecules is used to describe macroscopic parameters.

• One uses variables such as N (number of particles), v (velocity) to produce pressure.

KINETIC THEORY

• The microscopic movement of molecules is used to describe macroscopic parameters.

• One uses variables such as N (number of particles), v (velocity) to produce pressure.

• See the java applet which simulates this

• http://www.phy.ntnu.edu.tw/java/idealGas/idealGas.html

KINETIC THEORY

• The microscopic movement of molecules is used to describe macroscopic parameters.

• One uses variables such as N (number of particles), v (velocity) to produce pressure.

• See the java applet which simulates this• http://

www.phy.ntnu.edu.tw/java/idealGas/idealGas.html

• http://www.physics.org/Results/search.asp?q=Tell+me+about+kinetic+theory&uu=0

KINETIC THEORY

• When the molecules are in thermal equilibrium, then the average velocity in each direction is the same.

• < vx > = < vy > = < vz >

KINETIC THEORY

• When the molecules are in thermal equilibrium, then the average velocity in each direction is the same.

• < vx > = < vy > = < vz >

• The average velocity of all the molecules is < v > = 0 (since they are confined)

KINETIC THEORY

• When the molecules are in thermal equilibrium, then the average velocity in each direction is the same.

• < vx > = < vy > = < vz >

• The average velocity of all the molecules is < v > = 0 (since they are confined)

• Note the average speed is < v > ≠ 0 nor is the < v2 > ≠ 0 vRMS = √ (< v2 >

KINETIC THEORY

• When the molecules are in thermal equilibrium, then the average velocity in each direction is the same.

• < vx > = < vy > = < vz >

• The average velocity of all the molecules is < v > = 0 (since they are confined)

• Note the average speed is < v > ≠ 0 nor is the < v2 > ≠ 0 vRMS = √ (< v2 > and

< v2 > ≠ (< v >)2

KINETIC THEORY

• Let the internal energy be

• U = N<K>

where <K> is the average energy of one molecule.

KINETIC THEORY

• Let the internal energy be

• U = N<K>

where <K> is the average energy of one molecule. So U = N (½ m < v2 > )

KINETIC THEORY

• Let the internal energy be

• U = N<K>

where <K> is the average energy of one molecule. So U = N (½ m < v2 > )

but < v2 > = < vx 2 + vy 2 + vz 2 >

KINETIC THEORY

• Let the internal energy be

• U = N<K>

where <K> is the average energy of one molecule. So U = N (½ m < v2 > )

but < v2 > = < vx 2 + vy 2 + vz 2 >

and because U = U(T)

KINETIC THEORY

• Let the internal energy be

• U = N<K>

where <K> is the average energy of one molecule. So U = N (½ m < v2 > )

but < v2 > = < vx 2 + vy 2 + vz 2 >

and because U = U(T) then

< v2 > = < vx 2 > + < vy 2 > + < vz 2 >

KINETIC THEORY

• Since the gas is not moving then

< vx 2 > = < vy 2 > = < vz 2 >

KINETIC THEORY

• Since the gas is not moving then

< vx 2 > = < vy 2 > = < vz 2 >

therefore < v2 > = 3 < vx 2 >

KINETIC THEORY

• Since the gas is not moving then

< vx 2 > = < vy 2 > = < vz 2 >

therefore < v2 > = 3 < vx 2 > and

< vx 2 > = < v2 > /3

KINETIC THEORY

• Since the gas is not moving then

< vx 2 > = < vy 2 > = < vz 2 >

therefore < v2 > = 3 < vx 2 > and

< vx 2 > = < v2 > /3 = 2/3 (U/(mN))

KINETIC THEORY

• Since the gas is not moving then

< vx 2 > = < vy 2 > = < vz 2 >

therefore < v2 > = 3 < vx 2 > and

< vx 2 > = < v2 > /3 = 2/3 (U/(mN))

Let us now consider all the molecules in an element of cylindrical volume in the direction x dV = A vx dt

KINETIC THEORY

• In that volume let us consider collisions with those molecules having a velocity vx .

KINETIC THEORY

• In that volume let us consider collisions with those molecules having a velocity vx .

• The collisions will produce a change in momentum Δ PMOL = mvxf – mvxi = -2mvx .

KINETIC THEORY

• In that volume let us consider collisions with those molecules having a velocity vx .

• The collisions will produce a change in momentum Δ PMOL = mvxf – mvxi = -2mvx .

• The wall receives the reaction momentum

• Δ Pwall(x) = 2mvx

KINETIC THEORY

• In that volume let us consider collisions with those molecules having a velocity vx .

• The collisions will produce a change in momentum Δ PMOL = mvxf – mvxi = -2mvx .

• The wall receives the reaction momentum

• Δ Pwall(x) = 2mvx

• Since we are considering all the molecules with an x component velocity; vx is <vx> .

KINETIC THEORY

• In order to obtain the total momentum transferred to the wall in a time dt, one must know the number of collisions

KINETIC THEORY

• In order to obtain the total momentum transferred to the wall in a time dt, one must know the number of collisions

NCOLL = (N /V) dV = ½ (N/V) A vx dt

where ½ N/V is the number density of molecules, in the element of volume dV.

KINETIC THEORY

• In order to obtain the total momentum transferred to the wall in a time dt, one must know the number of collisions

NCOLL = (N /V) dV = ½ (N/V) A vx dt

where ½ N/V is the number density of molecules, in the element of volume dV.

The total x momentum transferred is

dPx = NCOLL ΔPwall(x)

KINETIC THEORY

• dPx = ½ (N/V) (A vx dt) (2mvx )

KINETIC THEORY

• dPx = ½ (N/V) (A vx dt) (2mvx )

thus in a time dt, the change in momentum is

dPx/dt = Fx = (N/V) m(vx)2 A

KINETIC THEORY

• dPx = ½ (N/V) (A vx dt) (2mvx )

thus in a time dt, the change in momentum is

dPx/dt = Fx = (N/V) m(vx)2 A

and the pressure is

P = Fx/ A = (N/V) m(vx)2

KINETIC THEORY

• dPx = ½ (N/V) (A vx dt) (2mvx )

thus in a time dt, the change in momentum is

dPx/dt = Fx = (N/V) m(vx)2 A

and the pressure is

P = Fx/ A = (N/V) m(vx)2

Since (vx)2 is an average over all molecules

P = (N/V) m<vx2>

KINETIC THEORY

• Since < vx 2 > = 2/3 (U/(mN))

KINETIC THEORY

• Since < vx 2 > = 2/3 (U/(mN)) then

P = (N/V) m 2/3 (U/(mN) = 2/3 (U/V)

KINETIC THEORY

• Since < vx 2 > = 2/3 (U/(mN)) then

P = (N/V) m 2/3 (U/(mN) = 2/3 (U/V)

or

PV = 2/3 U

KINETIC THEORY

• Since < vx 2 > = 2/3 (U/(mN)) then

P = (N/V) m 2/3 (U/(mN) = 2/3 (U/V)

or

PV = 2/3 U

Using the Ideal Gas Law PV = NkT

KINETIC THEORY

• Since < vx 2 > = 2/3 (U/(mN)) then

P = (N/V) m 2/3 (U/(mN) = 2/3 (U/V)

or

PV = 2/3 U

Using the Ideal Gas Law PV = NkT and

U =N<K>

KINETIC THEORY• Since < vx 2 > = 2/3 (U/(mN)) then P = (N/V) m 2/3 (U/(mN) = 2/3 (U/V)or PV = 2/3 U

Using the Ideal Gas Law PV = NkT and U =N<K> then kT = 2/3 U/N

KINETIC THEORY• Since < vx 2 > = 2/3 (U/(mN)) then

P = (N/V) m 2/3 (U/(mN) = 2/3 (U/V)

or

PV = 2/3 U

Using the Ideal Gas Law PV = NkT and

U =N<K>

then kT = 2/3 U/N = 2/3 <K>

or 3/2 kT = U/N = <K>