statistical mechanics

Question 1

How to find heat capacity

equipartition theorem

Answer 1

1. each T / R =1/2kT
2. each V = kT

Question 2

Heat capacity of monotonic

Answer 2

3* Trans =3/2kT

Question 3

Heat capacity of diatomic

Answer 3

3T + 2R + 1V

= 7/2kT

Question 4

How to find no. of modes of vibration

for equipartition

Answer 4

3N-6 for non-linear
3N-5 for linear

Question 5

Three modes of vibration

Answer 5

1. Assymetrical
2. Symetrical
3. Bending

Question 6

What does the boltzman’s distribution tells you

Answer 6

The ratio of population between two energy level

Question 7

How to find the population (in mols) at a specific energy level

if we know N, total population

partition function

Answer 7

1. use eq. - set j to ground state
2. hence find sum of population as a submation
3. then replace the N0 in the submation

Question 8

What can we analysis from the partition function

Answer 8

kT = currency of the molecule
ΔE = object to access

if kT needs to be sufficient to buy ΔE

Question 9

What are the most probable speed, mean speed, and root mean square speed?

Answer 9

most probable - largest population
mean speed - avg from the graph
Crms - from avg KE

Question 10

When does degeneracy in boltzman’s distribution apply

Answer 10

rotational energy level

Question 11

When is vibrational and rotational modes accessible

Answer 11

when temp. > threshold temp

Question 12

what does Cv,m and Cr,m
mean

Answer 12

vibration/rotational molar heat capacity contribution

Question 13

Describle Cv graph

Answer 13

1. at low temp, only translational motion
2. rotational then vibration modes are accessible as it passes each threshold temp
3. high temp may then cause molecule to dissociate

Question 14

How to find heat capacity

Answer 14

1. find the no. of modes (T,R,V)
2. estimate the θv and θr
3. if T > θr = find rotational Cm,r
4. if T> θv = find vibrational Cm,v
5. add up all three contributional

(compare with equapartition theoerm-depending on mode accessible)

Question 15

Assumptions of
kinetic theory of gas

Answer 15

1. translation is not quantised
2. KE is conserved
   (elastic collision)
3. rigid, point masses
4. random infrequent motion

Question 16

Consider 1D - derive

Maxwell boltzman distrubution

along a 1 dimension

Answer 16

1. E of molecule = KE
2. integrate the partition func. using gaussian integral
3. complete for all 3 dimension
4. cube root for solution

Question 17

What does the maxwell-boltzman distribution show

Answer 17

The probability of a molecule at a certain Crms/energy

Crms = root mean sq speed

Question 18

How to find avg. KE

from maxwell-boltzman

Answer 18

via rms speed

1. find mean sq speed from boltzman function
   [integrate v × f(v) , proability]
   +require gaussian integral
2. then use 1/2 mv^2

Question 19

Define collision rate

Answer 19

no. of collision per sec.

Question 20

Define mean free path

Answer 20

Avg. dist. travelled before collision

Question 21

How to derive diffusivity/viscosity, thermal conductivity

Answer 21

1. Using eq of flux
2. set up three windows at -l, 0, l
3. find flux at -l and l
4. do balance to relate to flux eq. (gradient of quantity)

Question 22

how to find quantity transfered from flux

Answer 22

flux * time * area of window

Question 23

Derivation of ideal gas eq.

from kinetic theory model

Answer 23

1. find force, i.e. change in momentum in collision
   (assume elastic)
2. find pressure from area of wall
3. use the definiton of rms speed from M-B distribution

Question 24

When does ideal gas eq. fails

Answer 24

At VLE, i.e. phase change

Question 25

What are the 2 modified assumption of the VDW eq.

Answer 25

1. point masses -> finite size sphere 2. introduce potential energy

Question 26

What can we find from the maxwell contruction

Answer 26

VLE line = line bounding the two Maxwell loop with equal area

Question 27

how to find the most favoured seperation from the Lennard-Jones potential

Answer 27

find dV/dr = 0 (where the potential is minimum)

Question 28

what is g(r) | Intermolecular potential

Answer 28

the likelihood to finding a particle at different seperation

Question 29

What happen to g(r) when V increase

Answer 29

More WD needed to reach seperation less likely to find molecule hence dec.

Question 30

4 step developing Lennard-Jones g(r)

Answer 30

1. for ideal gas - potential is same everywhere 2. but now infinite potential when r< radius of sphere 3. now include attractive and repulsive forces (weakens over seperation) 4. line needs to be continuous

Question 31

Explain the g(r) of solid

Answer 31

1. kT cannot access the energy well 2. can only be at the position of lowest potential 3. hence fixed positions

Question 32

Explain g(r) for gas

Answer 32

1. energy well is << kT 2. not quantised 3. potential is same everywhere

Question 33

Explain g(r) of liquid

Answer 33

1. kT around energy well 2. can occasionally overcome hence +local order +closely packed

Question 34

Explain activation energy in terms of energy well

Answer 34

Energy required to jump from one well to another

Question 35

How does temperature affect diffusion and viscosity

Answer 35

As temp inc. 1. Diffusion inc. 2. viscosity dec.

Question 36

Define polytypes

Answer 36

Identical 2D structure differ in 3D

Question 37

2 rules for solid crystaline structure

Answer 37

1. fill up as much as possible 2. overall charge = 0

Question 38

Define coordination number

Answer 38

No. of neighbours each molecule has

Question 39

packing order expression

Answer 39

vol. taken by particle/vol. of cube

Question 40

How to find heat capacity of solid lattice ? (using Boltzman's distribution)

Answer 40

**particle can only oscillate on the spot** 1. ΔE = 1/2ℏω 2. form partiton function 3. sum of E,energy × N,population = U, internal energy 3. total energy = inf. series (x^n = 1/1-x) 4.Cv = dU/DT

Question 41

What does point imperfection mean | in crystal defects

Answer 41

maybe: 1. missing atom 2. chemcial impurity

Question 43

Why does the viscosity of liquid decrease with temperature?

Answer 43

For liquid: inc. kT to overcome attraction For liquid: inc. collision between molecules

Question 44

Unit cell

Answer 44

smallest repeatable unit of the lattice

Question 45

Primitive cell

Answer 45

smallest possible unit of unit cell - containing one lattice point (min. volume)

Question 46

What is the difference between simple, body centred, face-centred

Answer 46

simple -containing a quarter each body-centred (bcc) - a particle in the centre face centred - no particle in the centre but each has face has a half on

Question 47

hat was the weakness of Einstein’s model and how did Debye improve this?

Answer 47

Einstein - assumed all vibrate at one frq fails at low temp Debye considers a spectrum of frq.

Question 48

What effects does defects have on crystal

Answer 48

May lead to: 1. reduced strength, conductivity 2. increased diffusion coefficient

Question 49

Why is L-J not suitable for solid ionic lattice

Answer 49

1. as prescence of electrostatic force, attraction and repulsion present between different molecule 2. instead a geometric series of potential generated instead

Question 50

What is the boltzman's constant, k

Answer 50

k = R/Na | Na = avogadro constant = no. of molecules per mol

Question 51

How to find number of molecules

Answer 51

1 mol = Na no. of molecules N= n* Na | N = no. of molecules Na = avogadro constant n = moles