Collision theory — frequency factor, steric factor

2.8.9 · Chemistry › Chemical Kinetics

Overview

Collision theory reaction rates ko molecular level par explain karta hai: molecules ko react karne ke liye sufficient energy KE SAATH proper orientation mein bhi collide karna padta hai. Frequency factor (A) total collision frequency aur geometric constraints ko quantify karta hai, jabki steric factor (ρ) orientation requirement ko capture karta hai.

[!intuition] Core Idea

Ek lock aur key ki tarah socho. Tum key ko darwaze mein 1000 baar per second thok sakte ho (collision frequency), lekin jab tak:

1. Tum kaafi hard hit nahi karte (activation energy)
2. Key sahi orientation mein nahi hoti (steric factor)

...darwaza nahi khulegaa. Reactions bhi isi tarah kaam karti hain.

WHY does this matter? Even agar molecules kaafi energy ke saath collide karein, tab bhi zyaadatar collisions galat angles par "glancing blows" hoti hain. Sirf ek fraction ke paas hi purane bonds todne aur naye banane ki geometry hoti hai.

[!definition] Frequency Factor (A)

Arrhenius equation mein frequency factor (jise pre-exponential factor bhi kehte hain):

$k=A{e}^{-E_a/RT}$

A KYA hai? Maximum possible rate constant agar saare collisions mein kaafi energy hoti (T → ∞). Yeh combine karta hai:

• Collision frequency (Z) — molecules kitni baar per unit time/volume collide karte hain
• Steric factor (ρ) — correct orientation wale collisions ka fraction

$A=\rho Z$

Units: Rate constant k ke jaisa hi (bimolecular ke liye: M⁻¹s⁻¹, unimolecular ke liye: s⁻¹)

Typical values: Bimolecular gas reactions ke liye 10⁸ se 10¹³ (M⁻¹s⁻¹)

[!formula] Collision Frequency (Z) Derive Karna

Step 1: Kinetic Molecular Theory Starting Point

Radii r_A aur r_B wale spherical molecules A aur B ke liye:

Collision cross-section: ${\sigma }_{AB}=\pi (r_A+r_B{)}^2$

WHY? Collision tab hoti hai jab centers (r_A + r_B) distance ke andar aate hain. Molecule A ke around (r_A + r_B) radius ka ek circle imagine karo — koi bhi B center is circle mein ghusta hai toh collision hoti hai.

Step 2: Relative Velocity

Molecules randomly move karti hain. Average relative speed:

${\bar{v}}_{rel}=\sqrt{\frac{8k_{B}T}{\pi \mu }}$

jahan reduced mass $\mu =\frac{m_A{m}_B}{m_A+m_B}$

WHY relative velocity? Jo matter karta hai woh yeh hai ki molecules ek dusre ke paas kitni tezi se aati hain, unki individual speeds nahi. Agar dono same direction mein same speed se move karein, relative velocity zero hai — koi collision nahi.

WHY yeh formula? Maxwell-Boltzmann distribution se. Jab tum do velocity distributions combine karte ho, relative velocity distribution bhi Maxwell-Boltzmann hoti hai lekin reduced mass ke saath.

Step 3: Collision Frequency Z

Unit volume per unit time mein AB collisions ki number:

$Z_{AB}={\sigma }_{AB}{\bar{v}}_{rel}[A][B]=\pi (r_A+r_B{)}^2\sqrt{\frac{8k_{B}T}{\pi \mu }}[A][B]$

HOW padhen ise:

• Bade molecules (larger σ) → zyaada collisions
• Higher temperature (larger v̄) → zyaada collisions
• Higher concentrations → zyaada collisions

Identical molecules (A + A) ke liye:

$Z_{AA}=\frac{1}{2}{\sigma }_{AA}{\bar{v}}_{rel}[A{]}^2$

WHY 1/2? Double-counting avoid karne ke liye (A₁ ka A₂ se milna A₂ ka A₁ se milne wali same collision hai).

[!formula] Steric Factor (ρ)

$\rho =\frac{k_{observed}}{Z\cdot e^{-E_a/RT}}=\frac{\text{effective collisions}}{\text{energetically favorable collisions}}$

ρ KYA represent karta hai? Properly oriented collisions ka fraction.

Typical values:

• Simple atoms (jaise noble gases): ρ ≈ 1 (koi bhi orientation kaam karta hai)
• Small molecules (H₂ + I₂): ρ ≈ 0.1 se 0.5
• Complex molecules (proteins): ρ ≈ 10⁻⁶ se 10⁻⁴

WHY ρ < 1 hota hai?

Reaction A-B + C → A + B-C ke liye, C atom ko A-B ke B end par hit karna padega, A end par nahi. Agar molecule cylindrical hai aur reaction site surface ka 1/10 cover karti hai, toh ρ ≈ 0.1.

Complete Rate Expression Ki Derivation

Collision theory se starting:

$\text{Rate}=(\text{collision frequency})\times P(\text{energy}\ge E_a)\times P(\text{correct orientation})$

$\text{Rate}=Z_{AB}\cdot e^{-E_a/RT}\cdot \rho$

$=\rho Z_{AB}{e}^{-E_a/RT}[A][B]$

Rate law ke liye: Rate = k[A][B], hume milta hai:

$k=\rho Z_{AB}{e}^{-E_a/RT}$

Arrhenius se compare karte hue: $k=A{e}^{-E_a/RT}$

$\boxed{A=\rho Z_{AB}}$

[!example] Worked Example 1: H₂ + I₂ ke liye Z Calculate Karna

Given: T = 500 K, r_H₂ = 1.0 Å, r_I₂ = 2.0 Å, [H₂] = [I₂] = 0.1 M

Find: Collision frequency Z

Solution:

Step 1: Collision cross-section $\sigma =\pi (r_{H_2}+r_{I_2}{)}^2=\pi (1.0+2.0{)}^2\times 10^{-20}{\text{ m}}^2=2.83\times 10^{-19}{\text{ m}}^2$

WHY radii add karte hain? Collision tab hoti hai jab molecular surfaces touch karti hain.

Step 2: Reduced mass $\mu =\frac{m_{H_2}\cdot m_{I_2}}{m_{H_2}+m_{I_2}}=\frac{(2\text{ g/mol})(254\text{ g/mol})}{256\text{ g/mol}}\cdot \frac{1}{N_A}=3.32\times 10^{-24}\text{ g}$

WHY reduced mass? Yeh dono molecules ke movement ko account karta hai.

Step 3: Relative velocity ${\bar{v}}_{rel}=\sqrt{\frac{8k_{B}T}{\pi \mu }}=\sqrt{\frac{8(1.38\times 10^{-23})(500)}{3.14(3.32\times 10^{-27})}}=631\text{ m/s}$

WHY T ke saath increase hota hai? Higher temperature → tezi se molecular motion.

Step 4: Collision frequency $Z=\sigma {\bar{v}}_{rel}[H_2][I_2]$

Concentrations convert karo: [H₂] = 0.1 M = 0.1 mol/L = 6.02 × 10²⁵ molecules/m³

$Z=(2.83\times 10^{-19})(631)(6.02\times 10^{25}{)}^2=6.5\times 10^{32}{\text{ collisions/(m}}^3\text{⋅s)}$

Physical meaning: Ek cubic meter mein, H₂ aur I₂ molecules 6.5 × 10³² baar per second collide karti hain.

[!example] Worked Example 2: Experiment Se Steric Factor Nikalna

Given: NO + O₃ → NO₂ + O₂ ke liye 298 K par:

• Observed rate constant: k = 1.8 × 10⁷ M⁻¹s⁻¹
• Activation energy: E_a = 10 kJ/mol
• Calculated collision frequency: Z = 5.0 × 10¹⁰ M⁻¹s⁻¹

Find: Steric factor ρ

Solution:

Step 1: Boltzmann factor calculate karo $e^{-E_a/RT}=e^{-10000/(8.314\times 298)}=e^{-4.03}=0.018$

WHY J use karte hain, kJ nahi? R = 8.314 J/(mol·K), units match karni chahiye.

Step 2: Collision theory apply karo $k=\rho Z{e}^{-E_a/RT}$

$\rho =\frac{k}{Z{e}^{-E_a/RT}}=\frac{1.8\times 10^7}{(5.0\times 10^{10})(0.018)}=\frac{1.8\times 10^7}{9.0\times 10^8}=0.02$

Interpretation: Sufficient energy wale sirf 2% collisions ki correct orientation hoti hai.

WHY itna low hai? NO ko O₃ ke paas N end ke saath approach karna padta hai (O end se nahi). Iske alawa, approach angle ko electron transfer ke liye orbital overlap allow karna padta hai.

[!example] Worked Example 3: A Ki Temperature Dependence

Question: Kya frequency factor A temperature par depend karta hai?

Answer: Thoda sa YES, Z ke through.

$Z\propto \sqrt{T}$ se, hume milta hai $A=\rho Z\propto \sqrt{T}$

Quantitative check:

$A(T_2)=A(T_1)\sqrt{\frac{T_2}{T_1}}$

300 K → 600 K (T double karne par):

$\frac{A(600)}{A(300)}=\sqrt{2}=1.41$

WHY yeh matter karta hai? Exponential term $e^{-E_a/RT}$ is range mein 10⁴ se 10⁶ ke factors se change hota hai, toh √T change negligible hai. Hum A ko constant treat karte hain Arrhenius plots mein.

Step-by-step reasoning:

• 300 K par: $e^{-E_a/RT}\approx 10^{-20}$ (typical E_a = 100 kJ/mol ke liye)
• 600 K par: $e^{-E_a/RT}\approx 10^{-10}$
• Exponential 10¹⁰ se increase hota hai
• A 1.4 se increase hota hai
• Exponential completely dominate karta hai

[!mistake] Common Misconception: "A Sirf Collision Frequency Hai"

Galat idea: A ek second mein collisions ki number ke equal hai.

Kyun sahi lagta hai: "Frequency factor" naam se lagta hai ki yeh sirf collision counting hai.

Kyun galat hai: A mein DONO collision frequency (Z) aur steric factor (ρ) include hote hain. Zyaadatar collisions fail hoti hain kyunki molecules galat oriented hoti hain.

Fix: $A=\underset{\text{orientation}}{\underbrace{\rho }}\times \underset{\text{collision frequency}}{\underbrace{Z}}$

Evidence: Complex molecules ki reaction ke liye, A, Z se 10⁶ times chhota ho sakta hai. Agar A sirf Z hota, toh saare "kaafi hard" collisions react kar lete — lekin nahi karte.

Steel-manning the mistake: Early collision theory (1920s) mein A ko Z ke barabar maana jaata tha kyunki unhone simple atomic reactions study ki thi jahan ρ ≈ 1 tha. Ar + Ar ke liye, yeh actually kaam karta hai! Error tab aaya jab chemists ne ise molecular reactions par apply kiya.

[!mistake] ρ Ko Activation Energy Ki Probability Se Confuse Karna

Galat idea: Steric factor account karta hai un molecules ke liye jinka E ≥ E_a hai.

Kyun sahi lagta hai: Dono "fractions" hain jo reaction rate reduce karte hain.

Kyun galat hai: Boltzmann factor $e^{-E_a/RT}$ energy ko pehle se handle karta hai. Steric factor ρ ADDITIONAL hai — yeh un collisions ke subset par apply hota hai jinmein pehle se kaafi energy hai.

Correct picture:

1. Total collisions: Z[A][B]
2. Energy filter ke baad: $Z[A][B]\cdot e^{-E_a/RT}$
3. Orientation filter ke baad: $Z[A][B]\cdot e^{-E_a/RT}\cdot \rho$

Yeh multiply karte hain, overlap nahi karte.

Test: Agar ρ energy account karta, toh T badhane se k exponentially nahi badhta — lekin badhta hai! Exponential temperature dependence prove karta hai ki energy aur orientation alag-alag filters hain.

[!recall]- 12-Saal-Ke-Bachhe Ko Explain Karo

Imagine karo tum ek video game khel rahe ho jahan tum balls fenkte ho targets par unhe todne ke liye.

Collision frequency (Z): Tum per minute kitni balls fenkte ho. Agar tum tezi se fenko ya zyaada balls ho, Z badhta hai.

Activation energy filter: Targets ke paas shields hain. Sirf kaafi tezi se phenki gayi balls (kaafi energy wali) hi andar ghus sakti hain. Temperature yeh hai ki tum kitna zor se phenko — higher T matlab zyaada zor se throw.

Steric factor (ρ): Even agar tum kaafi zor se phenko, tumhe red bullseye spot hit karna padega, blue edges nahi. Red spot sirf 10% target ho sakta hai. Toh ρ = 0.1 matlab har 10 "kaafi hard" throws mein se sirf 1 hi target ko actually todhti hai.

Frequency factor A = ρ × Z: Per minute maximum targets jo tum tod sakte the agar tum infinitely hard phenko (taaki shield matter na kare). Yeh combine karta hai tum kitni tezi se phenko (Z) aur teri aim kitni precise honi chahiye (ρ).

Chemistry mein, molecules balls hain, reactions targets todna hain, aur "red bullseye" woh exact angle hai jahan atoms partners swap kar sakte hain.

[!mnemonic] A = ρZ Yaad Karna

"A-ha! Rho Zones the action"

• A-ha: Frequency factor A
• Rho: Steric factor ρ (pehla letter)
• Zones: Collision frequency Z space ke zones mein hoti hai

Yaad rakho: "Proper Aim Requires Orientation" → A mein ρ orientation requirements se aata hai.

Typical values ke liye: "One Zero" → Simple molecules ke liye ρ ≈ 0.1 se 1.0, complex molecules 0.01 ya us se bhi neeche jaate hain (extra zeros).

Connections

• Arrhenius Equation: A pre-exponential factor hai
• Activation Energy: Sirf E ≥ E_a wali collisions react karti hain
• Maxwell-Boltzmann Distribution: Jahan se $e^{-E_a/RT}$ factor aata hai
• Transition State Theory: Zyaada sophisticated model jo A ko molecular vibrations se calculate karta hai
• Rate Laws: Microscopic collisions aur macroscopic rates ke beech connection
• Temperature Dependence of Reaction Rates: Kyun k T ke saath exponentially badhta hai
• Molecularity: Collision theory sirf elementary reactions par apply hoti hai

#flashcards/chemistry

Kaunse do factors satisfy hone chahiye taaki molecular collision reaction result kare? :: (1) Colliding molecules ki kinetic energy ≥ activation energy E_a honi chahiye (2) Colliding molecules ki proper orientation honi chahiye (steric factor ρ se capture hoti hai)

Arrhenius equation mein frequency factor A define karo :: Frequency factor A (pre-exponential factor) woh maximum rate constant hai agar saare collisions mein sufficient energy hoti. Yeh A = ρZ ke barabar hota hai, steric factor aur collision frequency ka product. Units k ke jaisi hain.

Molecular parameters ke terms mein bimolecular reaction A + B ke liye collision frequency Z likho :: Z_AB = σ_AB × v̄_rel × [A][B], jahan σ = π(r_A + r_B)² collision cross-section hai aur v̄_rel = √(8k_BT/πμ) average relative velocity hai. μ reduced mass hai.

A ko temperature-independent kyun treat karte hain jabki Z∝ √T hai? :: Halanki A = ρZ, √T ke saath badhta hai, yeh effect negligible hai. T double karne par A √2 ≈ 1.4 factor se badhta hai, jabki e^(-E_a/RT) 10⁴ se 10⁶ ke factors se badhta hai. Exponential term completely dominate karta hai.

Concept Map