Equilibrium constant from E° - ln K = nFE° - RT

2.7.6 · Chemistry › Redox & Electrochemistry (Intro)

Yeh Relationship Humein Kya Batati Hai

Equilibrium constant K tab products aur reactants ka ratio measure karta hai jab reaction change karna band kar deti hai. Standard cell potential E° voltage measure karta hai jab saari species 1 M, 1 bar, 25°C par hain. Yeh alag-alag worlds lagte hain, lekin yeh Gibbs free energy ΔG° ke through connected hain:

$Δ G ° = - n F E ° = - R T ln K$

jahan:

n = reaction unit per transfer hone wale electrons ki sankhya (moles of e⁻)
F = Faraday constant ≈ 96,485 C/mol
R = gas constant = 8.314 J/(mol·K)
T = absolute temperature (Kelvin)

First Principles se Derivation

Voltage aur equilibrium connect kyun hote hain? Dono same energy landscape describe karte hain—bas alag units mein.

Step 1: Gibbs Free Energy as the Master Function

Kisi bhi process ke liye, Gibbs free energy ΔG ka change spontaneity determine karta hai:

ΔG < 0 → spontaneous (forward)
ΔG = 0 → equilibrium (koi net change nahi)
ΔG > 0 → non-spontaneous (reverse favored)

Equilibrium par, ΔG = 0, jo equilibrium constant define karta hai:

$Δ G = Δ G ° + R T ln Q$

Equilibrium par, Q = K aur ΔG = 0:

$0 = Δ G ° + R T ln K$

$Δ G ° = - R T ln K ..(1)$

Yeh step kyun? Hum equilibrium ko thermodynamic language mein express kar rahe hain. ΔG° reaction ki "standard energy price" hai.

Step 2: Electrical Work ko Free Energy se Connect Karna

Ek electrochemical cell mein, chemical energy electrical work mein convert hoti hai. n moles electrons ke liye jo potential difference E° se move karte hain, maximum electrical work hai:

$w_{elec} = - n F E °$

Negative kyun? Thermodynamics mein system dwara surroundings par kiya gaya work negative hota hai. Ek spontaneous cell work karta hai, isliye w < 0.

Constant T aur P par, maximum work free energy change ke barabar hota hai:

$Δ G ° = - n F E ° ...(2)$

Yeh step kyun? Yeh electrochemistry-thermodynamics dictionary hai: voltage (E°) energy per coulomb mein translate hoti hai, aur nF total charge moved deta hai.

Step 3: Equate Karo aur Solve Karo

Equations (1) aur (2) se:

$- R T ln K = - n F E °$

$R T ln K = n F E °$

$ln K = \frac{n F E °}{R T}$

Yeh kyun matter karta hai: Ab hum measurable voltage se K predict kar sakte hain, ya equilibrium data se E° calculate kar sakte hain.

Figure — Equilibrium constant from E° - ln K = nFE° - RT

Worked Examples

Worked example Example 1: Silver-Zinc Cell

Diya gaya: $Zn (s) + 2 Ag^{+} (a q) \to Zn^{2 +} (a q) + 2 Ag (s)$ $E °_{cell} = E °_{Ag^{+} / Ag} - E °_{Zn^{2 +} / Zn} = 0.80 - (- 0.76) = 1.56 V$

Dhundho: 25°C par Equilibrium constant K.

Solution:

Step 1: n identify karo (transfer hone wale electrons)

Zn → Zn²⁺ + 2e⁻
2Ag⁺ + 2e⁻ → 2Ag
==n = 2==

Yeh step kyun? Electrons ki sankhya overall reaction mein balance honi chahiye. Har Ag⁺ ek electron leta hai, do Ag⁺ ko do electrons chahiye, jo ek Zn atom provide karta hai.

Step 2: 298 K par ln K = nFE°/RT apply karo $ln K = \frac{2 \times 96485 \times 1.56}{8.314 \times 298.15}$

$ln K = \frac{301 , 033.2}{2 , 478.9} = 121.4$

Yeh step kyun? Humari derived formula mein direct substitution. Bahut bada numerator (charge × voltage) vs. chhota denominator (thermal energy) ek massive K signal karta hai.

Step 3: K ke liye solve karo

$K = e^{121.4} \approx 1 0^{52.7}$

Interpretation: K astronomically large hai—yeh reaction essentially completion tak jaati hai. Equilibrium par, [Zn²⁺]/[Ag⁺]² ≈ 10⁵³. Positive E° (1.56 V) ka matlab hai ki reaction energy release karti hai, isliye products dominate karte hain.

Common mistake Common Error: Multi-Electron Transfers mein n Bhool Jaana

Galat soch: "Formula ln K = FE°/RT hai, bas E° plug in karo."

Yeh sahi kyun lagta hai: Students formulas yaad karte hain bina units track kiye. F ki units C/mol hain, E° J/C hai, isliye FE° J/mol hai—energy jaisa lagta hai.

The steel-man: Actually, FE° energy per mole of electrons hai. Lekin reaction 2 ya 3 electrons per formula unit transfer kar sakti hai. Agar aap n bhool jaate hain, toh aap alag reaction stoichiometry ke liye K calculate karte hain.

Fix: Balanced half-reactions mein hamesha electrons count karo. $Zn + Cu^{2 +} \to Zn^{2 +} + Cu$ ke liye, n = 2 (1 nahi). Agar aap n = 1 use karte hain, toh aapko K ki jagah K¹/² milta hai.

Khud check karo: Agar E° = 1.1 V hai aur aap K calculate karte hain, phir n double karo—kya K roughly square hota hai? Hona chahiye. ln K, n ke saath linearly scale karta hai.

Common mistake Common Error: E° ke Liye Non-Standard Concentrations Use Karna

Galat soch: "Mere cell mein [Cu²⁺] = 0.5 M hai, toh main woh ln K = nFE/RT mein use karunga."

Yeh sahi kyun lagta hai: Nernst equation actual concentrations use karti hai, isliye students isko equilibrium formula se mix up kar dete hain.

The steel-man: Aap concentrations aur voltage ko connect kar rahe hain, lekin equilibrium formula specifically standard potential E° (sab 1 M) ko K se relate karta hai. Agar aap arbitrary concentrations par measured non-standard E use karte hain, toh formula toot jaata hai.

Fix: ln K = nFE°/RT ko E° (standard conditions) chahiye. Agar aapke paas non-standard conditions par measured Ecell hai, pehle Nernst use karke E° find karo:

$E_{cell} = E ° - \frac{R T}{n F} ln Q$ E° ke liye solve karo, phir K calculate karo.

Analogy: E° ek "sea level" reference jaisa hai. Aap K ko mountain par khade hokar (non-standard conditions) aur woh altitude directly use karke measure nahi kar sakte.

Physical Intuition: Voltage Equilibrium Kyun Predict Karta Hai

E° ko reaction ki driving force ka measure samjho. Positive E° ka matlab hai ki electrons anode se cathode ki taraf "flow karna chahte" hain—energy release hoti hai. Equilibrium par, forward aur reverse reactions balance ho jaati hain, lekin products aur reactants ka ratio batata hai ki balance hone se pehle reaction kitni door gayi.

Large positive E° → bahut badi energy release → reaction almost completion tak jaati hai → K ≫ 1
E° ≈ 0 → driving force nahi ke barabar → equilibrium par products aur reactants similar → K ≈ 1
Negative E° → forward push karne ke liye energy chahiye → reverse direction favored → K < 1

Formula ln K = nFE°/RT bas ek equation hai, lekin physically: jitna bada voltage, utna exponentially bada K banta hai. 25°C par har 0.059 V increase (n = 1 ke liye) K ko 10 se multiply karta hai.

Recall Ek 12-Saal-Ke Bacche Ko Explain Karo

Ek water slide imagine karo. Slide ki height (jaise E°) batati hai ki neeche slide karne se aapko kitni energy milti hai. Agar slide bahut oonchi hai, toh aap bahut tezi se neeche aate hain aur shuruat se bahut door pahunch jaate hain—yeh bade K jaisa hai (bahut saare products). Agar slide flat hai, toh aap muskil se hilte hain—chhota K (zyaadatar reactants bache). Aur agar slide upar ki taraf jaati hai, toh aap bina kisi ke dhakka diye slide hi nahi kar sakte—yeh negative E° hai, K less than 1.

Formula ln K = nFE°/RT ek calculator jaisa hai jo slide ki height (E°) leta hai aur exactly batata hai ki aap kahan pahunchenge (K). n yeh hai ki kitne dost saath mein neeche aate hain (zyaada dost = zyaada total energy), aur RT friction jaisa hai—zyaada temperature = zyaada random jiggling = thoda sa change karta hai ki aap kahan rukenge.

Connections

Nernst Equation – non-standard conditions par E describe karta hai; equilibrium par, E = 0 aur Q = K
Gibbs Free Energy and Spontaneity – ΔG° = -nFE° = -RT ln K teeno ko unify karta hai
Standard Reduction Potentials – E°cell, E°cathode - E°anode tables se aata hai
Le Chatelier's Principle – bada K ka matlab hai ki reactants add karne se equilibrium muskil se shift hota hai
Thermodynamics of Electrochemical Cells – heat, work, aur voltage ko relate karta hai
Faraday's Laws of Electrolysis – charge (nF) ko moles of substance se connect karta hai
Arrhenius Equation – similar exponential form: k = Ae^(-Ea/RT)

#flashcards/chemistry

Positive E° value K ke baare mein kya batata hai?

K > 1 (equilibrium par products favored). Positive E° ka matlab ΔG° < 0 hai, isliye ln K > 0, thus K > 1.

ΔG° se ln K aur E° ke beech relationship derive karo.

ΔG° = -RT ln K (equilibrium condition) aur ΔG° = -nFE° (electrochemical work). Equate karte hue: -RT ln K = -nFE°, isliye ln K = nFE°/RT.

Reaction 2Fe³⁺ + Cu → 2Fe²⁺ + Cu²⁺ ke liye jisme E° = -0.43 V aur n = 2, 25°C par K estimate karo.

log K = nE°/0.0592 = (2)(-0.43)/0.0592 ≈ -14.5, isliye K ≈ 10⁻¹⁴·⁵ ≈ 3 × 10⁻¹⁵ (reactants strongly favored).

Formula mein n (moles of electrons) kyun include hai?

Kyunki nF reaction ke per mole mein transfer hone wala total charge deta hai. Zyaada electrons transfer = zyaada electrical work = ΔG° aur thus K par zyaada impact.

25°C par, E° ko 0.059 V badhane par (n=1 ke saath) K kitne factor se multiply hota hai?

10×. Kyunki log K = E°/0.0592, E° mein 0.059 V add karne par log K mein ~1 add hota hai, jo K ko 10 se multiply karta hai.

Agar K = 1, toh E° kya hai?

E° = 0 V. Jab K = 1, ln K = 0, isliye nFE°/RT = 0, thus E° = 0.

Denominator (RT) mein T physically kya represent karta hai?

Thermal energy. Zyaada T ka matlab zyaada random molecular motion, jo equilibrium distribution ko "smear out" karta hai, ek given E° ke ln K impact ko reduce karta hai.