Stat Thermo Exam 1

in a microcanonical ensemble BLANK stays constant across system copies

Energy, Volume, Number of particles

In a microcanonical ensemble, system copies M are not necessarily in the same

state

In a canonical ensemble BLANK stays constant across system copies, but BLANK is exchanged

Volume and Number of particles, Energy

When energy is exchanged in a canonical ensemble, this is called

fluctuation

In a grand canonical ensemble BLANK is constant across copies of the system, but BLANK is exchanged

Volume, Energy and Number of particles/mass

The statistical average of a thermodynamic dependent ensemble variable is determined by

the sum of the variable across all copies, divided by the total number of copies M

Equal apriori probability states that

all states with the same energy are equally likely, and in a closed system with fixed E, V, and N, all states are equally likely

Entropy grows with

degeneracy

S(E,V,N) =

klnΩ(E,V,N)

In thermodynamic equations, k is

the boltzmann constant

k is equivalent to

R/N_A

The degeneracy of M systems Ω_M is equal to

Ω^M

E (particle in a box) =

(h²/8mL²)(n_x²+n_y²+n_z²), where L² = V^3/2

dE =

TdS - pdV + μdN

dS =

(1/T)dE + (p/T)dV - (μ/T)dN

=

1/T

in determining E from S, E =

(3/2)NkT

= BLANK, where BLANK is equal to BLANK

P/T, where P/T is equal to Nk(1/V)

C_v=

=(3/2)Nk

n_i

the number of members with energy E_i

{n_i}

set of occupation numbers

for n_i, when i gets large all higher n_i

are equal to zero

Weith (W) =

N!/(n₀!n₁!n₂!…), where N is total number of partipating particles, and n_i is the number of particles participating in that specific state

the total degeneracy of a system, Ω_M =

the sum of all W

dlnW =

0

Q (canonical partition function) =

Σ_ie^-βE_i

P_i =

n_i/M = e^-βE_i/Q

Average E per copy can also be determined by the

product of E_i and P_i

Average energy E =

S (average entropy) =

kβE + klnQ

β =

1/kT

P_i =

A(T,V,N) =

-kTlnQ = E - TS

in terms of average energies, C_v =

the degeneracy of a rigid rotor is

(2J+1)

for a harmonic approximation, the molecular partition function (degeneracy dependent) q =

1/(1-e^-βε)

Ξ (grand canonical partition function) =

Σ_N e^-γNQ(N), where Q is the canonical partition function, and γ = -μ/kT

in terms of the grand canonical partition function, E =

where γ = -μ/kT

in terms of the grand canonical partition function, N =

entropy S of a grand canonical ensemble =

μ =

= -kTγ

grand canonical ideal gas law

pV = kTlnΞ

λ (generalized activity) =

e^μ/kT

in terms of activity, Ξ =

average N in terms of grand canonical partition =

the probability of finding a particle in any state with N particles =

how different a copy N is to average N is equal to

σ_N/N (where σ_N is the square root of the root mean square)

σ_N² =

κ_T (isothermal compressibiity) does not change with

increasing number of systems

κ_T is NOT well behaved

near a phase transition

for an ideal gas, κ_T =

1/p

For 2 gaseous independent sub-systems we assume that V_AB is

far less in magnitude to H_A+H_B

wavefunctions are BLANK while energies are BLANK

multiplicative, additive

Thermodynamic functions are

additive

lnΞ =

λq

q_T (where T indicates translation energy only) =

Λ (in units of length) =

Λ is called the

characteristic thermal wavelength

if q » N and Q_N = q^N/N!, then

V/Λ³ » N or NΛ³ « 1

ΔS for isothermal expansion =

Nkln(V₂/V₁)

in isothermal expansion, as the volume drops, e-levels available for translation

become closer together until they smooth to no longer be quantized

in an adiabatic process, the gas law appears as

pV^γ = constant where γ = C_p/C_V

γ is called

the adiabatic index

for monoatomic gases in an adiabatic process, ΔS =

0 (using the derivates with respect to helmholtz A)

What is the relationship between partition functions (canonical Q) in an adiabatic process

Q₂ = Q₁

what is the classical value of the adiabatic index for monoatomic gases

5/3

linear degrees of freedom

3n - 5

non-linear degrees of freedom

3n - 6

q sums over BLANK not BLANK

states, levels

Q for polyatomic gases appears as

q^N/N! = (q_T^N/N!)q_R^Nq_V^Nq_el^N

C_p - C_V =

R (the gas constant)

In a potential well the lowest possible value of E_T is BLANK, E_R is BLANK, E_vib is BLANK, E_el is BLANK, and lowest possible energy E is BLANK

0, 0, 1/2ћω, -D_e, -D₀

in a potential well D_e =

D₀ + 1/2ћω

D₀ is the

necessary energy to dissociate a molecule

D_e is the

depth of the potential energy well

rotational energies ε_i =

BJ(J+1), where B = ћ²/2I

q_rot =

OR T/Θ_R

Θ_R =

B/K

Θ_R is called

the characteristic rotational temperature

when is q_rot = T/Θ_R true?

when T » Θ_R

ε_vib =

q_vib =

A_vib =

Θ_vib =

ћω/k

Θ_vib is called the

characteristic vibrational temperature

if T » Θ_vib, q_vib =

if T » Θ_vib,

Θ_vib/T = ћω/Tk « 1

near T = 0 K,

C_v ∝ T³

ε as T → 0 =

ћω/2

ε as T → ∞ =

ћω/2 +kT

dS =

dq_rev/T

Extensive thermodynamic properties

E, V, n, S, n_i

Intensive thermodynamic properties

p, T, μ_i

in terms of E, T =

in terms of E, -p =

in terms of E, μ_i =

H =

E + pV

dH =

A =

E - TS

dA =