Lecture 5

Question 1

What is the formula for internal energy for WEAKLY COUPLED AND INDEPENDENT molecules?

Answer 1

U (T) = U (0) - N (del ln(q)/del B) V

U (T) is internal energy at a given temp
U(0) is the internal energy at ground state

Question 2

What is the formula for internal energy for a system which is INDEPENDENT BUT INTERMOLECULAR FORCES ARE ACCOUNTED FOR?

Answer 2

U (T) = U(0) - (del ln(Q)/del B) v

Question 3

What is the only mode of vibration for monoatomic gases?

Answer 3

Translational

Question 4

What is the equation for internal energy for monoatomic gases?

Answer 4

U (T) = U(0) + N(E)

E is mean energy

This is then used with the mean energy for translational which is
[E] = 3/2kT

So this is then U (T) = U (0) + 3/2NkT

Question 5

How do we get the molar internal energy?

Answer 5

Set N - Na
U(T) = U(0) + 3/2NakT
and because Nak = R
U(T) = U(0) + 3/2RT

Question 6

How is the heat capacity calculated? (constant volume)

Answer 6

Cv = deltaU/deltaT
or
Cv = (del(U)/del(T))V

The internal energy is now substituted in to get
Cv = d/dt (Um(0)+3/2RT)
Cv = 3/2R

Question 7

What does the heat capacity NOT rely on?

Answer 7

Atomic weight

Question 8

What is the heat capacity of a monoatomic gas?

Answer 8

Cv = 3/2R
Cv = 3/2 x 8.314
Cv = 12.471 J mol-1 K-1

THIS IS AT A CONSTANT VOLUME

Question 9

What are heat capacities more commonly reported as?

Answer 9

Constant pressure.

C(p,m) - C(v,m) = R

So, C(p,m) = 5/2R

Question 10

What is the vibronic contribution too CONSTANT-VOLUME heat capacity?

Answer 10

Related by Cv = dU/dT

Using the molar internal energy and the vibrational mean energy.

0v = hcv/k
[Ev] = k0^v/e^(0v/T) -1

The internal energy and heat capacity can be wrote as an expression. The partial derivative can then be taken of the new equation
Cv = (delNa[Ev)/delT)V

Mean vibrational energy is now subbed in, then the derivative is taken to give.

Cv = R(0^v/T)^2 e^(0^v/T)/(e^(0^v/T)-1)^2

Question 11

How else can heat capacity be written?

Answer 11

Taking the derivative of T with respect to B, = -kB^2(d/dB)

Subbing this in for heat capacity
Cv = -kB^2 (delU/delB)V

Now using internal energy (only the N[E] part and [E] = -(dellnq/delB)

Gives Cv = -NkB^2 (dellnq/delB^2)V

Question 12

What is the stat mech entropy formula?

Answer 12

S = klnW

in J K-1

Question 13

What happens to entropy as temp decreases?

Answer 13

Fewer states accessible, entropy and W decreases.

Question 14

What is the third law of thermodynamics?

Answer 14

States that the entropy of all perfect crystalline substances is zero at T=0.

Question 15

What happens as T->0?

Answer 15

Only 1 state, E=0. So W =1, S = kln1 = 0

Question 16

What is it called when a crystal can exist in more than one energetically equivalent state at 0K then S(0)>0?

Answer 16

Residual entropy

Question 17

What is the first example of a solid with residual entropy?

Answer 17

Ice

Question 18

What is the structure of ice? and hence the residual entropy?

Answer 18

1 oxygen joined to 4 H’s.
The oxygen forms too short sigma bonds too 2 H’s
and two long hydrogen bonds.

There are 2^2N possible arrangements of the 2N hydrogen atoms in NH2O.

Not all of these are acceptable. Only 6 of the 16 ways have 2 short and 2 long bonds.

The number of permitted arrangements is W = 2^2N(6/16)^N

Product rule gives 2^2N(6/16)^N = 4N x (6/16)^N = (3/2)^N.

So, the residual entropy for ice is; kNln(3/2)
N is substituted for Na for molar entropy
k x Na = R
Rln(3/2) = 3.4 J mol-1 K-1

Which is the exact experimental value.