12.5 Kinetic Model of Temperature (equations for pressure of an ideal gas on the walls of container and internal energy of an ideal gas)

Question 1

How do we derive the equation for the pressure exerted by the gas on a wall?

Answer 1

• work out change in momentum for the particle colliding with a wall in the x direction
• work out time for that to happen
• work out the rate of change of momentum (force) (newtons second law)
• Apply Newton’s Third Law to work out equation for force on the wall
• work out pressure by putting above equation into p = F/a
• consider the equation for more than one particle by integrating the different velocities into the equation
• is the new velocity, where is the mean square speed velocity in that one direction
• now consider the 3 directions (x,y,z), assuming pressure is split equallt between them. hence integrate 1/3 into the equation

Question 2

When we use the newtons third law what happens?

Answer 2

The minus sign disappears due to force on the wall acting in the opposite direction

Question 3

Derive the equation pV = 1/3Nm

Answer 3

momentum of 1 molecule = mvx
change in momentum after collision with wall = -mvx-mvx = -2mvx
time to do so = 2L / vx
the rate of change of momentum is -2mvx /2L/vx
this makes the equation -2m(vx)^2/2L
the 2s cancel out leaving -m(vx)^2/L
apply newton’s third law = m(vx)^2/L is now the force on the wall
integrate this into the P = F/A creating P = m(vx)^2/L^3
making P = m(vx)^2/V
consider more than one particle making (vx)^2 = (vx1^2,vx2^2,vx3^2) = pV = Nm
the (vx)^2 now represents the mean squre speed
consider all directions (x,y,z) with pressure being equally split
hence pV = 1/3Nm (vx changes into c for the correct notation)

Question 4

What does each symbol stand for in the pV = 1/3Nm?

Answer 4

p = pressure 
V = volume 
c^2 is the mean square speed 
c = root mean square speed 
N = number of molecules

Question 5

Is the root mean square the same as the mean?

Answer 5

No

Question 6

what is Nm/V and what does this mean for the equation?

Answer 6

Nm/V is equal to the density

this makes the equation p = 1/3(row)

Question 7

How do we work out the mean square speed?

Answer 7

Square each value
add values together
divide by the number of values

Question 8

How do we work out the root mean square speed?

Answer 8

Square root of the mean square speed

Question 9

What is the root mean squre speed?

Answer 9

The typical speed of the gas molecule

Question 10

What is the internal energy of an ideal gas equal to?

Answer 10

The kinetic energy of the gas particles, since there is no potential energy.

Question 11

What is the equation for the kinetic energy of the gas particles?

Answer 11

1/2 m = 3/2kT

Question 12

How do we derive this equation?

Answer 12

pv =NkT   and   pV = 1/3Nm 
1/3Nm = NkT 
Ns cancel out  
1/3m = kT 
divide by 2/3 
1/2m = 3/2kT 
hence equation for the average kinetic energy PER MOLECULE is 1/2m = 3/2kT

Question 13

When looking at above derived equation what is the proportional to what?

Answer 13

The average kinetic energy is proportional to the absolute temperature as k is a constant?

Question 14

In the equation 1/2m = 3/2kT what does each symbol stand for?

Answer 14

m = mass of molecule 
c = root mean square speed 
k = constant  (1.38x10^-23)
T = temperature in kelvin

Question 15

What is the equation for force exerted by the gas on 1 wall?

Answer 15

F = Nm(c^2) /L

Question 16

So a rise in temperature gives what?

Answer 16

A rise in Ek of each molecule and hence total internal energy

Question 17

What would happen if the temperature was at an absolute zero?

Answer 17

The molecules would have no kinetic energy

Question 18

What is a black body radiator?

Answer 18

body that absorbs all wavelengths of EM radiation and emits all wavelengths of EM radiation

Question 19

What is electromagnetic radiation emitted in?

Answer 19

discrete packets (quanta) called photons. This is called quantisation of energy

Question 20

Why do objects emit photons.

Answer 20

Due to their temperature. Most of the radiation lies in the infrared part of spectrum.