Topic 3

Question 1

What is an ideal gas?

Answer 1

A gas can be treated as an ideal gas if:

the volume occupied by the gas molecules is negligible compared to the volume occupied by the bulk gas (point particles)

the gas molecules are in random motion
all collisions can be treated as perfectly elastic

forces between different gas molecules can be neglected
i.e. there are no interactions between gas particles

Question 2

What is the heat capacity at constant volume equation?

Answer 2

𝐶_𝑉=1/2 𝑛𝑅𝑓

Question 3

Derive the equation for adiabatic expansion

Answer 3

𝑇𝑉^(𝛾−1)=constant 𝑝𝑉^𝛾=constant

𝛾≡𝐶_𝑝/𝐶_𝑉

Question 4

What is the net work done during full cycle of the Carnot cycle?

Answer 4

Area enclosed by path

Work done along each two isothermal paths and two adiabatic paths

Question 5

What is the efficiency of Carnot cycle?

Answer 5

𝜂=𝑊_𝑜𝑢𝑡/𝑄_𝑖𝑛 =1−𝑄_out/𝑄_in =1−𝑇_ℓ/𝑇_ℎ

Question 6

What is the efficiency of the Otto Cycle?

Answer 6

𝑊out/𝑄in=1−(𝑉_𝑚𝑎𝑥/𝑉_𝑚𝑖𝑛 )^((1−𝛾) )

Question 7

What is the work done equation in terms of equipartition (adiabatic)?

Answer 7

W=dU=fnR(Tf-Ti)/2 = W=(𝑝_𝑓 𝑉_𝑓 − 𝑝_𝑖 𝑉_𝑖)/(𝛾-1)

Question 8

What is the work done in isothermal compression/expansion?

Answer 8

W=-nRTln(Vf/Vi)

Question 9

What is the work done in adiabatic expansion/compression?

Answer 9

W=(𝑝_𝑓 𝑉_𝑓 − 𝑝_𝑖 𝑉_𝑖)/(𝛾−1)

Question 10

What is the mean collision time?

Answer 10

𝜏=1/(√(2⟨𝑣^2 ⟩ )4𝜋𝑟^2 𝑛)

where n= N/V = p/k_BT is the number of molecules per unit volume/number density

Question 11

What is the mean free path or the average distance travelled between collisions?

Answer 11

𝜆=1/(√2 4𝜋𝑟^2 𝑛)=(𝑘_B 𝑇)/(√2 𝑝𝜎);

where n is the number of molecules per unit volume/number density

We treat them as solid objects

Question 12

What is the collision cross section?

Answer 12

𝜎≡4𝜋𝑟^2

Question 13

What is the thermal conductivity of an ideal gas?

Answer 13

Determines how quickly heat is gained or lost

𝜅= 𝑘_𝐵 𝑓√(⟨𝑣^2 ⟩ ))/ 4√2( 4𝜋𝑟^2 ) = 1/4nk_Bf√(⟨𝑣^2 ⟩ ))𝜆
Units: Wm^-1K^-1

(Rate of (heat) thermal transfer Q=-𝜅AdT/dx)

Independent of pressure/density because in an ideal gas the only energy is the KE so the transfer of KE determines thermal conductivity

Question 14

What is Fick’s first law?

Answer 14

J=−𝐷 𝜕𝑐/𝜕𝑧
D - diffusion coefficient
Minus sign - arise since the net flow is in the opposite direction to the gradient
c- concentration

Question 15

What is Fick’s second law & continuity equation ?

Answer 15

𝜕𝑐/𝜕𝑡= -𝜕J/𝜕𝑧 (exact)
it is the change in concertation due to j of particles
c - concentration 
J - flux
dz - change in direction 

Combining Fick’s 1st law and continuity equation:

Making the assumption that flux is proportional to the gradient

In one dimension:
𝜕𝑐/𝜕𝑡=𝐷 (𝜕^2 𝑐)/(𝜕𝑧^2 )

(in three dimensions 𝜕𝑐/𝜕𝑡=𝐷𝛻^2 𝑐)

Question 16

What is Van der Waal equation of state/ How do you model a real gas ?

Answer 16

(𝑝+(𝑎𝑛^2)/𝑉^2 )(𝑉−𝑛𝑏)=𝑛𝑅𝑇

OR

(𝑝+(𝑎^′ 𝑁^2)/𝑉^2 )(𝑉−𝑁𝑏^′ )=𝑁𝑘_𝐵 𝑇

First term -Intermolecular forces
Second term - Volume of molecules

a(attractive in intermolecular forces) and b(atomic/molecular volume) characteristic of particular gas, determined by experiment

Note that it reduces to an ideal gas when V&raquo_space; nb, nsqrt(a)

Question 17

What is the critical isotherm?

Answer 17

𝑇_𝑐=(8𝑎^′)/(27𝑘_𝐵 𝑏^′ )

Question 18

What is the critical pressure?

Answer 18

𝑝_𝑐=𝑎^′/(27𝑏^′2 )

Question 19

What is the critical volume?

Answer 19

𝑉_𝑐=3𝑁𝑏′

Question 20

What is the Van der Waals equation of state in terms of scaled variables ?

Answer 20

𝑝̃=𝑝/𝑝_𝑐
𝑉̃=𝑉/𝑉_𝑐
𝑇̃=𝑇/𝑇_𝑐
(𝑝̃+3/𝑉̃^2)(3𝑉̃-1)=8𝑇̃

Question 21

What is the average speed of a gas molecule?

Answer 21

⟨𝑣^2 ⟩=fk_bT/m (from the equipartition theorem)

Question 22

How do you find the mass of a molecule?

Answer 22

m = Mr/N_a

Question 23

What is the efficiency of Carnot cycle?

Answer 23

η = Wout/Qin = 1-Qout/Qin = 1- Tl/Th

Question 24

What is the efficiency of the Otto cycle?

Answer 24

η = Wout/Qin=1-(Vmax/Vmin)^(1-gamma)

Question 25

What do we assume in the mean free path and thermal conductivity?

Answer 25

In microscopic derivation atomic radius represents the atom as a hard sphere

Not the case for the lighter elements

Question 26

What is diffusion?

Answer 26

Random and slow process. Process driven by diffusion is irreversible.

Question 27

What happens to the gases at high densities?

Answer 27

Ideal gas equation doesn’t hold

Increase density of gas it will start to condenses from gas to liquid which can not be described by the ideal gas