Heat Capacities

Question 1

Previous estimate:

Answer 1

3R for monatomic solids, all modes of motion contribute however we know this isn’t true as contribution for each mode varies with temperature

Question 2

Equipartition theorem with heat capacity:

Answer 2

Total internal energy has some contribution from translational, rotational and vibrational modes.

Question 3

Translational contribution and why

Answer 3

1/2 kT per mode, involves motion of one species

Question 4

Rotational contribution and why

Answer 4

1/2 kT per mode, involves motion of one species

Question 5

Vibrational contribution and why

Answer 5

kT per mode, involves motion of two species

Question 6

Vibrational activation temperature =

Answer 6

θ_v= hcṽ/k
ṽ is the wavenumber for that vibration

Question 7

Vibrational contribution becomes..

Answer 7

Long but in formula sheet

Question 8

Rotational activation temperature

Answer 8

θ_R = hcB/k
B is rotational constant

Question 9

Rotational contribution becomes

Answer 9

Long but in formula sheet

Question 10

Rotational activation temperature compared to vibrational activation temperature

Answer 10

Rotational is much lower so it’s contributions become fully activated at lower temperatures than vibrational

Question 11

Overall specific heat capacity

Answer 11

Cv = Cv_v +Cv_R + Cv_T

Question 12

Contributions from translation

Answer 12

Cv_T =3R/2 as activation temeperature is so low that it is considered to be fully activated

Question 13

What happens to the heat capacity at excessively high temperatures

Answer 13

Bonds between molecules will break (anharmonic) so molecule will be disassociated to a set of atoms that can only translate so Cv = N*3R/2