Entropy and Temperature

Question 1

What is the equation for entropy associated with a macrostate of an isolated system?

Answer 1

S = k(B)*ln(Ω), where Ω is the number of microstates in the macrostate.

Question 2

What can we learn from the equation for entropy?

Answer 2

That the more microstates accessible, the less defined the system is and the greater the entropy is -> entropy is related it disorder.

Question 3

What is S(total) equal to?

Answer 3

S(tot) = k(B)ln(Ω1Ω2) = k(B)ln(Ω1) + k(B)ln(Ω2) = S1 + S2

Question 4

What is isothermal expansion in thermodynamics?

Answer 4

2 systems connected with v0 in 1 and nothing in other at temp T, then goes to v0 in both with same T.

Question 5

What is the equation for ΔS in isothermal expansion?

Answer 5

ΔS = integral from initial to final of dS = integral from v0 to 2v0 of ρ dV/T = integral from v0 to 2v0 of Nk(B)/V dV (pV = Nk(B)T = Nk(B)ln(2)

Question 6

What happens in SM for the isothermal expansion problem?

Answer 6

Each molecules has 2 choices: LHS or RHS, so Ωtot = Ω0 * 2^N, where Ω0 is the original microstates, so ΔS = k(B)ln(2^N) = Nk(B)ln(2)

Question 7

What is the 2nd law in thermodynamics?

Answer 7

An isolated system in equilibrium is a state of maximum entropy.

Question 8

What is the 2nd law in SM?

Answer 8

An isolated system in equilibrium will adopt the state with the highest number of microstates.

Question 9

When is equilibrium reached in terms of microstates?

Answer 9

When Ωtot is max i.i d(Ω1Ω2)/dE1 = 0, where E1 is the energy in the first system. and Ω1 is the microstates in the first system etc

Question 10

What is the equation for dE2/dE1 in the 2 systems?

Answer 10

dE1/dE2 = -1

Question 11

What equation can we infer from the relationship between energies in the 2 systems?

Answer 11

dln(Ω1)/dE1 = dln(Ω2)/dE2 at thermal equilibrium.

Question 12

If we define T using 1/k(B)T = dln(Ω)/dE, what do we find?

Answer 12

That T1 = T2 at thermal equilibrium.

Question 13

In thermodynamics, what is the equation for 1/T?

Answer 13

1/T = (dS/dE)at V for an isolated system (V fixed)

Question 14

In SM, what is the equation for 1/T?

Answer 14

1/T = d(k(B)ln(Ω))/dE = dS/dE for fixed V

Question 15

What symbol do we assign to 1/k(B)T and what are its units?

Answer 15

1/k(B)T = β [J]

Question 16

What is a good diagram to show the canonical ensemble?

Answer 16

system with energy Ei (fixed T,N,V) which can exchange energy with reservoir at temperature T.

Question 17

What is a good example of the canonical ensemble?

Answer 17

1 SHO in contact with 2 SHOs, so Etot = 2ћω and Ωtot = 6

Question 18

What is the probability P(Ei) of the system being in energy state Ei?

Answer 18

P(2) = 1/6 = Ω(E-2)/6, P(1) = 2/6 = Ω(E-1)/6, P(0) = 3/6 = Ω(E-0)/6, so P(Ei) is proportional to Ω(E-Ei)

Question 19

How do we expand ln(Ω(E-Ei))?

Answer 19

ln(Ω(E-Ei)) = ln(Ω(E)) - dlnΩ(E))/dE *Ei = ln(Ω(E)) - Ei/k(B)T

Question 20

What do we get for the equation for Ω(E-Ei) and what does this mean?

Answer 20

Ω(E-Ei) = Ω(E)*exp(-Ei/k(B)T), meaning P(Ei) is proportional to exp(-Ei/k(B)T): The Boltzmann Distribution.

Question 21

What do we get from the normalization of P(Ei)?

Answer 21

Sum over i of P(Ei) = sum over i of exp(-βEi)/Z = 1, so Z = sum over i of exp(-βEi) -> the Partition function

Question 22

What is the chemical reaction example for the probability the reaction occurs?

Answer 22

P(T) proportional to exp(-βEa), where Ea is the activation energy, so can divide one probability by the other to get how much more likely one reaction is than the other.

Question 23

For a 2 state system at level 0 and level E, what does equal?

Answer 23

= sum over i of EiPi = 0(P(0) + EP(E) = Eexp(-βE)/Z, so Z = exp(-β0)+exp(-βE), so = E*exp(-βE)/(1+exp(-βE)) = E/(exp(βE)+1)