MELT

Equipartition principle

Each squared term in the expression for the energy contributes ½ RT to the molar internal energy of the system, Um,v

Translational contribution to internal energy

3/2 RT (x, y, z)

Rotational contribution to U m,v (non-linear)

3/2 RT

Rotational contribution to U m,v (linear)

RT

Vibrational contribution to U m,v

RT

Order of energies from highest to lowest

Electronic, vibrational, rotational, translational

Relationship between C and U

C = (dU/dT)v

Relationship between U and q

U = (N Kb T² * dq/dT) / q

Usual meaning of N

number of particles, moles = N / NA

U m =

U / n

Helmholtz energy, A, =

U - TS

State

Unique energy eigenfunction of the system

Level

One of the quantized energy levels that the system may occupy

Equation to find total microstates

(E units + N distinguishable particles - 1) C (N - 1)

Partition function of the system

Qn = ∑ exp(-Ei/kT)

molecular partition function

q = ∑ gi exp(-e/kT)

relation between system and molecular partition function (distinguishable particles)

Qn = q^n

relation between system and molecular partition function (indistinguishable particles)

Qn = q^n / n!

what is q in words

the number of accessible states

what do you compare the energy to

kT

when calculate q term by term

if energy level spacing is comparable or larger than kT / only a few states can be occupied

dq/dt=

1/kT² ∑e exp(-e/kT)

degeneracy of rigid rotor

2J + 1

total q of a system

q(trans) x q(rot) x q(vib) x q(ele)

when k = 0.695

units = cm -1, like during rotational stuff

equation to get ‘k’ = 0.695

k / hc

when to use k=0.695

units in cm -1

what is ƒₙ

volume independent partition function

volume independent translational partition function

[2πmkT / h²]^(3/2)

partition functions available to electron

translational, electronic

do you consider the mass of an electron when a species gains an electron (M + e → M⁻) in the mass of M⁻?

No (M and M⁻ are roughly equal in mass)

for ƒ₁ / ƒ₂, what is the ratio of rotational contribution reduced to?

r²₁ / r²₂ (from reduced mass and B∼)

q(rot) =

T / (σ θr), σ = 1 (heternuclear) or 2 (homonuclear)

θr =

B∼ / 0.695 (hcB∼ / k_b)

q (vib) when GS = 0

1 / 1 - exp(-θv / T)

q (vib) when considering GS E

exp(-θv / 2T) / 1 - exp(-θv / T)

θv =

ω∼ / 0.695 (hcω∼ / k_b)

common q(elec) =

g₀ (degeneracy)

degeneracy of Π (term symbol)

2 (2 × 1 as can occupy both +1 and -1)

K(double dagger) =

exp(-∆G/RT)

∆G = (K)

-RT ln K

k_2nd=

(kT/h) (1/c) K(dd)

needed for volume of activation proof

d(∆G (dd)) / dp = ∆V(dd)

A in terms of Qn

A = -kT ln(Qn)

A in terms of U, T, S

A = U - TS

canonical probability distribution, Pm

Pm = 1/Qn exp(-Em/kT)

S in terms of microstates, w

S = k ln w

^NC_n

N! / n!(N-n)!

how to calculate number of particles, n, that occupy N sites

^NC_n

is the choose function additive or multiplicative

multiplicative

things to consider when thinking about entropy for adsorption

• mixing of particles

• distribution of particles

µ in terms of q

µₙ = εₙ⁰ - kT ln (qₙ / Nₙ)

∆G in terms of µ

∆G = Σn_iµ_i (products = +ve n, reactants = -ve n)

N for q°

N_A

partition functions of atoms

translational, electronic

V°

RT / P°

units of ∆ε in rate constant

cm-1, so k = 0.695