Collision theory — frequency factor, steric factor: Question bank

2.8.9 · D5 · Chemistry › Chemical Kinetics › Collision theory — frequency factor, steric factor

Ye bank parent note ke ideas drill karta hai: frequency factor and steric factor, aur ye Activation Energy, Maxwell-Boltzmann Distribution, Temperature Dependence of Reaction Rates, aur Transition State Theory par bhi rely karta hai.

Poora toolkit ek hi page par (quiz se pehle padho)

Traps se pehle, yahan neeche use hone wala har symbol, plain words mein define kiya gaya hai aur ek picture se joda gaya hai. Is page par koi bhi symbol aisa nahi hoga jo tumne pehle yahan dekha na ho.

Definition Core equation aur uske letters

Reaction ki speed ek master formula (Arrhenius equation) se capture hoti hai: $k = A e^{- E_{a} / R T}$

$k$ = rate constant: reaction ka "speed dial". Jitna bada $k$ , utni tez reaction. Iske units rate law se match karte hain: $s^{- 1}$ (unimolecular) ya $M^{- 1} s^{- 1}$ (bimolecular).
$A$ = frequency factor: reaction kitni tez hoti agar energy kabhi problem na hoti. Ye do cheezein se bani hai (neeche dekho).
$E_{a}$ = activation energy: wo minimum collision energy jo ek pair ko react karne ke liye chahiye — Figure s01 mein hill ki height.
$R$ = gas constant $= 8.314 J mol^{- 1} K^{- 1}$ : temperature aur molecular energy ke beech fixed conversion factor.
$T$ = absolute temperature kelvin mein: jitna zyada hot, utne tez molecules.
$e^{- E_{a} / R T}$ = Boltzmann factor: collisions ka wo fraction jo hill clear karne ke liye kaafi energetic hai (Figure s02 mein shaded tail).

Formula Collision frequency

$Z$ kahan se aati hai Molecule A ko ek gas of molecule B mein sweep karte hue imagine karo (Figure s04). Collision tab hoti hai jab unke centres $r_{A} + r_{B}$ ke andar aa jaate hain, jahan $r_{A}, r_{B}$ molecular radii hain (har sphere ki size). Toh har A ek "target disc" carry karta hai: $σ = π (r_{A} + r_{B})^{2} (the ==collision cross-section==)$ Time $t$ mein, ek A jo relative speed $\overset{v}{ˉ}_{r e l}$ par move kar raha hai, $σ \overset{v}{ˉ}_{r e l} t$ volume ki ek tube sweep karta hai. Ise volume per kitne B hain aur volume per kitne A hain se multiply karne par collision rate milti hai: $Z = Z_{unit} σ \overset{v}{ˉ}_{r e l} [A] [B]$ Exactly yahi reason hai ki $Z$ , $[A] [B]$ ke saath scale karta hai: dono mein se kisi bhi reactant ka zyada hona matlab hai ki har second mein zyada targets sweep honge. Concentration dependence yahan rehti hai; rate constant $k = ρ Z_{unit} e^{- E_{a} / R T}$ mein $[A] [B]$ divide out ho chuka hai.

Formula

$\overset{v}{ˉ}_{r e l}$ mein $μ$ kyun hai, aur $A \propto T$ kyun hai $\overset{v}{ˉ}_{r e l}$ mean relative speed hai — do molecules average par ek doosre ki taraf kitni tez approach karte hain. Maxwell-Boltzmann Distribution ko relative motion par apply karne se: $\overset{v}{ˉ}_{r e l} = \frac{8 k _{B} T}{π μ}$ Yahan $μ$ reduced mass hai. Reduced mass kyun? Ek two-body approach (dono molecules move kar rahe hain) ko ek effective particle jo mass $μ$ ki hai, ek fixed point ki taraf move karte hue re-write kiya ja sakta hai — ye standard two-body-to-one-body trick hai, toh do masses ek single $μ$ mein collapse ho jaate hain: $μ = \frac{m _{A} m _{B}}{m _{A} + m _{B}}$ Notice karo ki $μ$ hamesha kisi bhi ek mass se chota hota hai: product ko (bade) sum se divide karne par wo shrink ho jaata hai — physically, ye pair dono lumps mein se kisi bhi ek se zyada nimbly respond karta hai.

$T$ chain (ye missing derivation hai): $Z_{unit} = σ \overset{v}{ˉ}_{r e l}, \overset{v}{ˉ}_{r e l} \propto \frac{T}{μ} \Rightarrow Z_{unit} \propto T$ Kyunki $A = ρ Z_{unit}$ aur $ρ, σ$ $T$ par depend nahi karte, hume milta hai: $A \propto T$ Figure s02 dikhata hai ki ye weak $T$ growth exponential ke saamne kyun dub jaati hai.

Neeche har figure ko quiz items se pehle reference kiya gaya hai — jawab dene se pehle ek nazar daal lo.

True ya false — justify karo

Har claim ya toh subtly true hai ya subtly false. Sirf verdict mat do, kyun batao.

Frequency factor

A

ek second mein collisions ki number ke barabar hota hai.

False.

A = ρ Z_{unit}

— ye collision frequency

Z_{unit}

hai jo orientation ke liye steric factor

ρ

se scale down ki gayi hai. Sirf tab jab

ρ \approx 1

(colliding bare atoms) tab

A \approx Z_{unit}

Boltzmann factor

e^{- E_{a} / R T}

mein orientation requirement already include hai.

False.

e^{- E_{a} / R T}

purely wo fraction hai jiske paas kaafi energy hai (Figure s02 mein shaded tail). Orientation ek alag, multiplicative filter

ρ

hai — ye dono kabhi overlap nahi karte.

Agar kisi reaction ka zero activation energy hai, toh har collision reaction lead karti hai.

False.

E_{a} = 0

e^{- E_{a} / R T} = 1

ho jaata hai (energy filter sab pass kar deta hai), lekin orientation filter

ρ

phir bhi apply hota hai — molecules galat angle par bhi hit kar sakte hain jab tak

ρ = 1

bhi na ho.

Bade molecules hamesha tez react karte hain kyunki wo zyada baar collide karte hain.

False. Wo do collide karte hain zyada (bada cross-section

σ = π (r_{A} + r_{B})^{2}

), lekin unke paas mis-oriented hone ke bhi zyada tarike hote hain, toh

ρ

sharply drop karta hai — net rate aksar girr jaati hai.

Steric factor

ρ

kabhi 1 se zyada nahi ho sakta.

Saamanya mein False. Simple collision theory expect karta hai

ρ \leq 1

, lekin kuch real reactions (jaise harpoon/long-range electron transfer, Figure s03 right) mein

ρ > 1

dikhta hai — yeh sign hai ki hard-sphere model true reactive cross-section ko underestimate karta hai.

Identical-molecule reaction

A + A

ke liye, hum

Z

ko 2 se divide karte hain.

True.

\frac{1}{2}

factor double-counting rokta hai: "molecule 1 hits molecule 2" aur "molecule 2 hits molecule 1" ek hi collision hai.

Absolute temperature ko double karne se

A

ki value roughly double ho jaati hai.

False. Kyunki

A \propto T

T

double karne se

A

sirf

2 \approx 1.41

se badhta hai, matlab sirf lagbhag 41%.

T

ke saath dramatic rate increase exponential term se aata hai,

A

se nahi (Figure s02).

A

ke units hamesha

s^{- 1}

hote hain.

False.

A

ke wahi units hote hain jo rate constant

k

ke hote hain: unimolecular ke liye

s^{- 1}

, bimolecular ke liye

M^{- 1} s^{- 1}

(dekho Rate Laws).

Collision frequency ko individual velocity nahi, relative velocity govern karti hai.

True. Jo matter karta hai wo hai do molecules kitni tez approach karte hain; do molecules jo equal speed par side-by-side travel kar rahe hain unki relative velocity zero hoti hai aur wo kabhi collide nahi karte, isliye

\overset{v}{ˉ}_{r e l}

mein reduced mass

μ

appear karta hai.

Galti dhundho

Har statement mein ek flaw hai. Use name karo aur correct karo.

k = A e^{+ E_{a} / R T}

, toh zyada

E_{a}

se badi rate milegi."

Sign galat hai: ye

e^{- E_{a} / R T}

hota hai. Ek bada barrier matlab hai ek zyada negative exponent (Figure s01 mein ek unchi hill), toh kam collisions use clear kar paati hain aur

k

decrease karta hai.

"Boltzmann factor nikalne ke liye main

E_{a}

kJ mein

e^{- E_{a} / R T}

mein

R = 8.314

ke saath plug karta hoon."

Unit mismatch.

R = 8.314 J mol^{- 1} K^{- 1}

hai, toh

E_{a}

J/mol mein hona chahiye (

R T

se divide karne se pehle kJ ko 1000 se multiply karo).

ρ = k / (Z e^{- E_{a} / R T})

ka value 2000 aaya, toh orientation 2000 guna help karta hai."

Itna bada value ek modelling error flag karta hai, koi real orientation boost nahi — likely

Z

bahut chota compute kiya gaya tha ya

E_{a}

bahut zyada. Hard spheres ke liye genuine steric factors 1 par ya us se neeche hote hain.

"Collision cross-section

σ = π r_{A} r_{B}

use karta hai."

Galat combination. Ye

σ = π (r_{A} + r_{B})^{2}

hai — collision tab hoti hai jab centres radii ke sum ke andar aate hain (Figure s04), toh wo sum (squared) target-disc ka radius hai.

"Kyunki

A

mein

ρ

hai, aur

ρ \leq 1

A

hamesha

Z_{unit}

se chota hoga."

Aksar true, lekin "hamesha" nahi: jab

ρ > 1

(long-range reactive encounters) tab

A

Z_{unit}

se zyada ho jaata hai, jo hard-sphere picture ki limit expose karta hai.

"Reduced mass

μ = m_{A} + m_{B}

Ye total mass hai. Reduced mass hai

μ = \frac{m _{A} m _{B}}{m _{A} + m _{B}}

, jo hamesha kisi bhi individual mass se chota hota hai.

"Arrhenius plot of

ln k

1/ T

mein, slope

A

deta hai."

Slope

- E_{a} / R

deta hai; intercept (

1/ T \to 0

)

ln A

deta hai (Figure s05). Slope energy hai, intercept frequency factor hai.

Why questions

Sirf fact nahi, mechanism batao.

Complex molecules (jaise proteins) ke beech reactions ke liye steric factor

1 0^{- 6}

jitna chota kyun hota hai?

Har bade molecule ka sirf ek chota patch reactive site hota hai (Figure s03 left); zyataar collisions inert regions par hoti hain, toh correctly aligned hits ka fraction collapse ho jaata hai.

\overset{v}{ˉ}_{r e l}

mein ek molecule ki mass ki jagah reduced mass

μ

kyun use karte hain?

Kyunki dono partners move karte hain; two-body approach ko ek effective particle of mass

μ

ki tarah re-cast kiya jaata hai jo ek fixed point ki taraf move karta hai, toh do masses

μ

mein collapse ho jaate hain aur relative speed ki Maxwell-Boltzmann Distribution use karta hai.

Ordinary Arrhenius analysis mein hum

A

ko temperature-independent kyun treat karte hain?

Iska

T

variation exponential ke saamne trivial hai, jo usi range mein

1 0^{4}

–

1 0^{6}

ke factors se swing kar sakta hai (Figure s02) — toh

A

ka change noise mein kho jaata hai.

Collision theory molecular reactions ke liye rates kyun systematically over-predict karta hai?

Ye har energetic collision ko reactive count karta hai (

ρ = 1

), geometry ignore karke; real molecules ko specific alignment chahiye, toh measured rates prediction se neeche girti hain — ye gap hi steric factor hai. Transition State Theory ne baad mein

ρ

ko first principles se explain kiya.

Do noble-gas atoms ke beech reaction ke liye orientation irrelevant kyun hai?

Ek sphere ka koi "wrong end" nahi hota — koi bhi approach direction equivalent hai, toh

ρ \approx 1

aur

A \approx Z_{unit}

; exactly isliye early collision theory atomic reactions ke liye kaam karti thi.

Concentration badhane se collision frequency

Z

toh badhti hai lekin rate constant

k

kyun nahi?

Z = Z_{unit} [A] [B]

concentrations ke saath scale karta hai, lekin

k = ρ Z_{unit} e^{- E_{a} / R T}

per unit concentration define hota hai; concentration effects rate law

Rate = k [A] [B]

mein rehte hain,

k

mein nahi.

Low

ρ

ke liye kabhi kabhi sirf kaunsa end nahi, balki approach angle bhi kyun constrained hona chahiye?

Sahi atom ko hit karne ke alawa, colliding species ko reacting orbitals ko overlap karne dena chahiye; sahi site par ek glancing hit jo poor orbital alignment ke saath ho wo bhi fail ho jaati hai, jo

ρ

ko aur shrink karti hai.

Edge cases

Wo limits aur degenerate scenarios jo formulas quietly assume karte hain.

T \to \infty

hone par

e^{- E_{a} / R T}

ka kya hota hai?

1

ki taraf tend karta hai: ab har collision energy barrier clear kar leti hai, toh

k \to ρ Z_{unit} = A

. Isliye

A

ko "maximum possible rate constant" kehte hain (Figure s02, curve

A

par flatten ho jaata hai).

T \to 0

hone par

e^{- E_{a} / R T}

ka kya hota hai?

0

ki taraf tend karta hai: essentially koi bhi molecule ke paas needed energy nahi hoti, toh

k \to 0

aur reaction freeze out ho jaati hai — Temperature Dependence of Reaction Rates ke saath consistent.

Agar

E_{a} = 0

aur

ρ = 1

ho, toh

k

kya hoga?

k = Z_{unit}

— collision-limited maximum, kyunki dono filters sab kuch pass kar dete hain aur

A = ρ Z_{unit} = Z_{unit}

. Ye wo fastest hai jitna ek bimolecular reaction ja sakti hai (diffusion/collision limit).

ρ \to 1

physically colliding species ke baare mein kya demand karta hai?

Unhe kisi bhi mutual orientation se react karna chahiye — ye sirf structureless ya highly symmetric partners jaise single atoms ke liye true hai; ek specific reactive site wale molecules

ρ = 1

tak nahi pahunch sakte.

Agar computed

ρ

negative aaye, toh kya galat hua?

Physically impossible hai — negative fraction ek sign ya unit error signal karta hai (aksar exponent mein

E_{a}

sign ya kJ aur J ka mix), kabhi real reaction nahi.

Bahut heavy pair (bada

μ

) ke liye collision frequency ka kya hota hai?

\overset{v}{ˉ}_{r e l} = 8 k_{B} T / π μ

jaise

μ

badhta hai, girti hai, toh heavier partners dheeray move karte hain aur kam baar collide karte hain, orientation consider karne se pehle bhi

Z_{unit}

ko lower karti hai.

Recall Ek-line self-test

Upar ke har jawab ko cover karo aur un teen items ke liye reasoning (verdict nahi) re-derive karo jo tumhe sabse mushkil lagi. Agar tum kyunki explain kar sako, toh concept tumhara hai.