Statistical thermodynamics — partition functions, Q_trans, Q_rot, Q_vib, Q_elec; computing thermodynamic properties

5.1.5 · Chemistry › Physical Chemistry (Advanced)

1. Partition function kya hota hai?

YE form KYUN? Level $i$ occupy karne ki probability Boltzmann factor hai $p_{i} = g_{i} e^{- ε_{i} / k_{B} T} / q$ . Probabilities ka sum 1 hone ke liye ek normalizing constant chahiye — wahi constant hai $q$ . To $q$ literally temperature $T$ par effectively accessible states ko count karta hai.

$T \to 0$ par: sirf ground state matter karta hai, $q \to g_{0}$ (bas uski degeneracy).
$T \to \infty$ par: $e^{- ε_{i} / k_{B} T} \to 1$ sabhi levels ke liye, $q \to$ states ki number (bahut badi).

Molecular $q$ se system $Q$ tak

$N$ independent, indistinguishable molecules ke liye (ek ideal gas): $Q = \frac{q ^{N}}{N !}$ $N!$ un states ki overcounting hatata hai jo sirf identical molecules ko swap karne se alag dikhte hain.

Factorize KYUN? Agar total energy independent contributions ka sum hai, to Boltzmann exponential of a sum ek product ban jata hai — to partition functions multiply hote hain: $ε = ε_{t r an s} + ε_{r o t} + ε_{v ib} + ε_{e l ec} \Rightarrow q = q_{t r an s} q_{r o t} q_{v ib} q_{e l ec}$

Figure — Statistical thermodynamics — partition functions, Q_trans, Q_rot, Q_vib, Q_elec; computing thermodynamic properties

2. Master equations (ek baar derive karo, hamesha use karo)

$U$ KAISE milta hai (scratch se derivation): Mean energy probability-weighted sum hai: $U - U_{0} = ⟨ ε ⟩ N = N \frac{\sum _{i} ε _{i} e ^{- β ε_{i}}}{q} .$ Dhyan do $\sum_{i} ε_{i} e^{- β ε_{i}} = - \frac{\partial}{\partial β} \sum_{i} e^{- β ε_{i}} = - \frac{\partial q}{\partial β}$ . Isliye: $U - U_{0} = - \frac{N}{q} \frac{\partial q}{\partial β} = - N \frac{\partial l n q}{\partial β} = - \frac{\partial l n Q}{\partial β} .$ $\partial β \to \partial T$ convert karna $β = 1/ k_{B} T$ , $d β = - 1/ (k_{B} T^{2}) d T$ se $k_{B} T^{2} (\partial ln Q / \partial T)$ wala form deta hai. Yahi poora engine hai.

3. Chaar contributions

(a) Translational — $q_{t r an s}$

3-D box mein particle, energy spacings bahut chhoti → continuum ki tarah treat karo: $q_{t r an s} = \frac{V}{Λ ^{3}}, Λ = \frac{h}{2 π m k _{B} T}$ $Λ$ thermal de Broglie wavelength hai — molecule ki "quantum size."

$V / Λ^{3}$ KYUN? $q_{t r an s}$ count karta hai ki $Λ^{3}$ size ke kitne "quantum boxes" volume $V$ mein fit hote hain. Zyada jagah ya zyada $T$ (chhota $Λ$ ) → zyada accessible states.

Contribution: $U_{t r an s} = \frac{3}{2} N k_{B} T$ , $C_{V, t r an s} = \frac{3}{2} N k_{B}$ (equipartition: 3 translational DOF).

(b) Rotational — $q_{r o t}$ (linear molecule)

Levels $ε_{J} = h c B J (J + 1)$ , degeneracy $g_{J} = 2 J + 1$ , $B$ = rotational constant. Jab $k_{B} T ≫ h c B$ (room $T$ par usually), sum ko integral se replace karo: $q_{r o t} = \frac{k _{B} T}{σ h c B}$ $σ$ = symmetry number (HCl jaise heteronuclear ke liye =1, $N_{2}$ jaise homonuclear ke liye =2) — yeh indistinguishable rotated orientations ko remove karta hai.

Contribution: $U_{r o t} = N k_{B} T$ , $C_{V, r o t} = N k_{B}$ (linear ke liye 2 rotational DOF).

(c) Vibrational — $q_{v ib}$ (harmonic oscillator)

Levels $ε_{v} = (v + \frac{1}{2}) h ν$ . Ground state se measure karte hue, $ε_{v} = v h ν$ , ek geometric series: $q_{v ib} = \sum_{v = 0}^{\infty} e^{- v h ν / k_{B} T} = \frac{1}{1 - e ^{- h ν / k_{B} T}}$ Geometric KYUN? $\sum_{v = 0}^{\infty} x^{v} = 1/ (1 - x)$ jisme $x = e^{- h ν / k_{B} T}$ .

Vibrational temperature $θ_{v} = h ν / k_{B}$ define karo. Tab: $U_{v ib} = N k_{B} \frac{θ _{v}}{e ^{θ_{v} / T} - 1}, C_{V, v ib} = N k_{B} (\frac{θ _{v}}{T})^{2} \frac{e ^{θ_{v} / T}}{( e ^{θ_{v} / T} - 1 ) ^{2}} .$ High $T$ par: $q_{v ib} \to T / θ_{v}$ , $C_{V, v ib} \to N k_{B}$ (full equipartition). Low $T$ par: vibrations "freeze out" ho jaate hain, $C_{V, v ib} \to 0$ .

(d) Electronic — $q_{e l ec}$

Electronic gaps usually bahut bade hote hain ( $≫ k_{B} T$ ), to sirf ground state contribute karta hai: $q_{e l ec} = g_{0} (ground-state degeneracy)$ Zyaatar molecules ke liye $g_{0} = 1$ . Exceptions: O atom, NO (low-lying excited states), radicals.

4. Worked examples

5. Common mistakes (steel-manned)

6. Forecast-then-Verify

Recall Forecast:

$T \to \infty$ hone par, $C_{V, v ib}$ kya approach karta hai? Padhne se pehle predict karo. Verify: $C_{V, v ib} \to N k_{B}$ (per mode). High $T$ par discrete levels continuous lagte hain, equipartition restore karta hai $\frac{1}{2} k_{B}$ kinetic + $\frac{1}{2} k_{B}$ potential = $k_{B}$ per vibrational mode. ✓

Flashcards

Molecular partition function

q

ki definition

q = \sum_{i} g_{i} e^{- ε_{i} / k_{B} T}

— accessible states ka ek dimensionless Boltzmann-weighted count.

q_{t r an s}, q_{r o t}, q_{v ib}, q_{e l ec}

add kyun nahi hote, multiply kyun hote hain?

Energies add hoti hain, aur sum ka exponential ek product hota hai, isliye

q = \prod q_{k}

N

indistinguishable independent molecules ke liye system partition function

Q = q^{N} / N!

Q

se internal energy ka master formula

U - U_{0} = k_{B} T^{2} (\partial ln Q / \partial T)_{V} = - (\partial ln Q / \partial β)_{V}

Entropy ka master formula

S = (U - U_{0}) / T + k_{B} ln Q

q_{t r an s}

aur thermal wavelength

q_{t r an s} = V / Λ^{3}, Λ = h / 2 π m k_{B} T

Linear molecule ke liye

q_{r o t}

(high-

T

)

q_{r o t} = k_{B} T / (σ h c B)

σ

=symmetry number

Harmonic oscillator ke liye

q_{v ib}

q_{v ib} = 1/ (1 - e^{- h ν / k_{B} T})

(geometric series se)

q_{e l ec}

ki usual value

g_{0}

, ground-state degeneracy (aksar 1)

Room

T

par diatomic ke liye

C_{V, m}

\frac{5}{2} R

(trans

\frac{3}{2} R

+ rot

R

, vib frozen)

Symmetry number

σ

kis kaam aata hai?

Indistinguishable rotated orientations ki overcounting hatata hai (1 hetero, 2 homonuclear linear ke liye)

Sackur–Tetrode entropy

S_{m} = R [ln (V_{m} / N_{A} Λ^{3}) + \frac{5}{2}]

Recall Feynman: 12 saal ke bachche ko samjhao

Socho ek molecule ek building mein ek baccha hai jisme kaafi floors hain (energy levels). Jab thand hoti hai, baccha ground floor par rehta hai. Jab garam hota hai, baccha upar ke floors par daud sakta hai. Partition function bas yahi hai — "abhi kitne floors tak baccha aasani se pahunch sakta hai." Bada number matlab zyada floors kaam mein hain. Ek baar pata chal jaye ki kitne floors hain aur baccha usually kitna upar baithta hai, to sab kuch figure out kar sakte ho: total energy, kitne bachche spread out hain (entropy), pressure, sab kuch. Hum floors ko motion ke chaar types mein baante hain — slide karna (translation), ghoomna (rotation), thitharna/khichna (vibration), aur electron jumps (electronic) — har ek count karo, phir counts ko ek saath multiply karo.

Connections

Boltzmann distribution — weighting $e^{- ε / k_{B} T}$ ki origin
Equipartition theorem — har contribution ke high- $T$ limits
Ideal gas law — $p = k_{B} T (\partial ln Q / \partial V)_{T}$ se derive hota hai
Sackur–Tetrode equation — monatomic gases ki absolute entropy
Rotational and vibrational spectroscopy — $B$ aur $ν$ provide karta hai jo $q_{r o t}, q_{v ib}$ mein jaate hain
Heat capacity of gases — modes ke freeze-out se temperature dependence explain hoti hai