Nernst equation E = E° − (RT - nF) ln Q

2.7.4 · Chemistry › Redox & Electrochemistry (Intro)

Overview

Nernst equation electrochemical cell potential ko non-standard conditions mein standard cell potential aur reaction quotient se connect karta hai. Yeh thermodynamics aur electrochemistry ke beech ka bridge hai, jo dikhata hai ki concentration kaise voltage ko drive karta hai.

Yeh kyun important hai: Batteries standard conditions (1 M, 25°C, 1 atm) par operate nahi karti. Nernst equation humein batata hai ki ek real electrochemical cell mein aap actually kitna voltage measure karoge.

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[!intuition] Core Idea

Battery ko paani ke pahaad se neeche behne jaisi cheez samjho. "Unchaai" (voltage) do cheezein par depend karti hai:

1. Intrinsic height difference (E°) — reaction kitna chahta hai ki ho
2. Kitna "neeche" bacha hai — agar products pile up ho jaayein (high Q), toh driving force kam ho jaati hai

Jaise-jaise reaction aage badhti hai, reactants → products bante hain, Q badhta hai, aur voltage girata hai. Equilibrium par (Q = K), voltage zero ho jaata hai (driving force khatam). Nernst equation is decay ko quantify karta hai.

Physical meaning:

• High reactant concentration → badi driving force → zyada voltage
• High product concentration → reaction "satisfied" → kam voltage

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[!definition] Key Terms

• Standard cell potential (E°): Woh voltage jab saari species 1 M concentration, 1 atm pressure, aur 25°C (298 K) par hain. Yeh thermodynamic "starting height" hai.
• Cell potential (E): Kisi bhi condition mein actual voltage.
• Reaction quotient (Q): $Q=\frac{[\text{products}]}{[\text{reactants}]}$, kisi bhi moment par concentration ratio. Equilibrium par, Q = K.
• n: Balanced half-reactions mein transfer hone wale electrons ke moles ki sankhya.
• F: Faraday's constant = 96,485 C/mol, ek mole electrons ka charge.

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[!formula] First Principles se Derivation

Step 1: Gibbs Free Energy aur Cell Potential

Thermodynamics se shuru karte hain. Kisi bhi reaction ke liye driving force Gibbs free energy change hai:

$\Delta G=\Delta G°+RT\ln Q$

Yeh form kyun?

• $\Delta G°$ standard free energy hai (1 M, 1 atm).
• $RT\ln Q$ non-standard concentrations ke liye correction hai. Agar Q > 1 (zyada products), toh $\ln Q>0$, $\Delta G$ less negative (less spontaneous) ho jaata hai. Agar Q < 1 (zyada reactants), toh $\ln Q<0$, $\Delta G$ more negative (more spontaneous) ho jaata hai.

Step 2: ΔG ko Voltage se Link Karo

Electrochemical cell ke liye, electrical work jo hoti hai:

$\Delta G=-nFE$

Negative kyun? Ek spontaneous reaction ($\Delta G<0$) positive voltage produce karta hai (E > 0). Negative sign dono conventions ko align karta hai.

Isi tarah, standard conditions par:

$\Delta G°=-nFE°$

Step 3: Substitute Karo aur E Solve Karo

ΔG relations ko thermodynamic equation mein plug karo:

$-nFE=-nFE°+RT\ln Q$

Dono sides ko $-nF$ se divide karo:

$E=E°-\frac{RT}{nF}\ln Q$

Yahi Nernst equation hai.

Step 4: 25°C par Simplification

T = 298 K par, R = 8.314 J/(mol·K), F = 96,485 C/mol plug karo, aur ln ko log₁₀ mein convert karo (ln = 2.303 log₁₀):

$\frac{RT}{F}=\frac{8.314\times 298}{96,485}=0.0257\text{ V}$

$E=E°-\frac{0.0257}{n}\ln Q=E°-\frac{0.0592}{n}{\log }_{10}Q\text{(at 25°C)}$

log₁₀ kyun use karte hain? 10 ki powers mein concentrations ke saath mental math aasaan hoti hai.

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[!example] Worked Example 1: Zinc-Copper Cell

Reaction: $\text{Zn}(s)+{\text{Cu}}^{2+}(aq)\to {\text{Zn}}^{2+}(aq)+\text{Cu}(s)$

Diya hai: E° = 1.10 V, [Cu²⁺] = 0.01 M, [Zn²⁺] = 1.0 M, T = 25°C

Dhundhna hai: Actual cell potential E.

Step 1: Q likho.

$Q=\frac{[{\text{Zn}}^{2+}]}{[{\text{Cu}}^{2+}]}=\frac{1.0}{0.01}=100$

Yeh Q kyun? Solids (Zn, Cu) Q mein nahi aate. Sirf aqueous ions aate hain.

Step 2: n identify karo.

Zn → Zn²⁺ + 2e⁻ (oxidation) Cu²⁺ + 2e⁻ → Cu (reduction)

Toh ==n = 2==.

Step 3: Nernst equation apply karo.

$E=1.10-\frac{0.0592}{2}\log (100)$

$\log (100)=2$

$E=1.10-\frac{0.0592}{2}\times 2=1.10-0.0592=1.04\text{ V}$

E° se kam kyun? High [Zn²⁺] (product) driving force ko reduce karta hai.

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[!example] Worked Example 2: Concentration Cell

Setup: Do half-cells same electrode (Ag/Ag⁺) ke saath, lekin alag [Ag⁺].

Left: [Ag⁺] = 0.001 M Right: [Ag⁺] = 1.0 M

Dhundhna hai: Cell voltage.

Step 1: Recognize karo ki E° = 0.

Dono half-cells identical hain, toh standard potential difference zero hai. Voltage purely concentration gradient se aata hai.

Step 2: Spontaneous direction likho.

Electrons low [Ag⁺] se high [Ag⁺] ki taraf flow karte hain. Reaction:

${\text{Ag}}^{+}(1.0M)+e^{-}\to \text{Ag}(s)\text{(cathode)}$ $\text{Ag}(s)\to {\text{Ag}}^{+}(0.001M)+e^{-}\text{(anode)}$

Net: ${\text{Ag}}^{+}(1.0)\to {\text{Ag}}^{+}(0.001)$ — sirf dilution!

Step 3: Q likho.

$Q=\frac{[{\text{Ag}}^{+}{]}_{\text{anode}}}{[{\text{Ag}}^{+}{]}_{\text{cathode}}}=\frac{0.001}{1.0}=0.001$

Step 4: Nernst apply karo (n = 1).

$E=0-\frac{0.0592}{1}\log (0.001)=-0.0592\times (-3)=0.178\text{ V}$

Positive kyun? Reaction dilution direction mein spontaneous hai (high → low concentration). Q < 1 ln Q term ko negative banata hai, toh E > E° = 0 ho jaata hai.

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[!example] Worked Example 3: Equilibrium Par

Question: Jab cell equilibrium reach kar le tab E kya hoga?

Step 1: Equilibrium par, Q = K, aur ΔG = 0.

Step 2: ΔG = -nFE se, agar ΔG = 0, toh E = 0.

Step 3: E = 0 ko Nernst mein plug karo:

$0=E°-\frac{RT}{nF}\ln K$

Rearrange karo:

$E°=\frac{RT}{nF}\ln K\Rightarrow K=e^{nFE°/RT}$

25°C par:

$\log K=\frac{nE°}{0.0592}$

Matlab: Ek bada positive E° matlab ek bahut bada equilibrium constant (reaction almost completion tak jaati hai). E° aur K ek hi sikke ke do pahlu hain.

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[!mistake] Common Errors

Mistake 1: Q Mein Solids/Pure Liquids Ko Bhool Jaana

Galat soch: Q mein solid Zn ko [Zn] ki tarah include karna.

Kyun sahi lagta hai: "Zn reaction mein hai, toh Q mein hona chahiye."

Fix: Pure solid ya liquid ki activity = 1 by definition. Sirf gases (partial pressures) aur aqueous ions (concentrations) Q mein aate hain.

Sahi: $\text{Zn}(s)+{\text{Cu}}^{2+}\to {\text{Zn}}^{2+}+\text{Cu}(s)$ ke liye, $Q=\frac{[{\text{Zn}}^{2+}]}{[{\text{Cu}}^{2+}]}$.

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Mistake 2: Log Ki Jagah Ln Use Karna (Ya Vice Versa)

Galat: Dono Nernst forms ko mix karna.

Kyun hota hai: Do versions exist karte hain (ln aur log₁₀). 0.0592/n use karne ke liye log₁₀ chahiye. 0.0257/n use karne ke liye ln chahiye.

Fix:

• General form: $E=E°-\frac{0.0257}{n}\ln Q$ (natural log)
• 25°C, base-10: $E=E°-\frac{0.0592}{n}\log Q$

Ek choose karo aur ussi pe tikay raho. Agar ln Q diya hai, 0.0257 use karo. Agar log Q diya hai, 0.0592 use karo.

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Mistake 3: n Ka Galat Sign

Galat: Yeh sochna ki n negative ho sakta hai agar electrons doosri side par hain.

Kyun sahi lagta hai: "Agar electrons right side par hain, toh shayad n negative hai?"

Fix: n hamesha positive hota hai. Yeh electrons transfer ki count hai. Direction (oxidation vs. reduction) already E° ke sign mein baki hai jab tum overall cell reaction likhte ho.

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Mistake 4: E aur E° Ko Confuse Karna

Galat: Table se E° use karna jab problem non-standard concentrations de rahi ho.

Kyun hota hai: E° dhundhna aasaan hai, aur students Nernst apply karna bhool jaate hain.

Fix: E° sirf standard conditions par sahi hai. Agar concentrations ≠ 1 M, temperature ≠ 25°C, ya pressure ≠ 1 atm, toh Nernst zaroor use karo E dhundhne ke liye.

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[!mnemonic] Memory Aid

Nernst ke liye "NERD":

• Not standard? Use Nernst.
• E° starting voltage hai.
• RT/nF correction factor hai.
• Don't forget ln Q (ya log Q at 25°C).

Concentration intuition: "Zyada reactants → zyada voltage. Zyada products → kam voltage."

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[!recall]- Feynman Explanation (Age 12)

Socho tumhare paas ek battery hai jo alag-alag unchaai par rakhe do paani ke buckets se bani hai. Upar wala bucket (reactants) neeche wale bucket (products) mein baehna chahta hai, aur woh behna ek waterwheel ko power karta hai (voltage generate karta hai).

Ab, E° woh original height difference hai jab dono buckets bhari hain. Lekin jaise paani behta hai, upar wala bucket thoda khali hota hai, aur neeche wala bharta hai. Height difference kam hoti hai, toh waterwheel dheere ghumti hai (voltage girta hai).

Nernst equation tumhe batata hai: "Kitna voltage bacha hai?" Yeh depend karta hai:

1. Starting height par (E°).
2. Abhi har bucket mein kitna paani hai (Q, concentration ratio).

Agar upar wala bucket (reactants) almost khali hai aur neeche wala (products) almost bhar gaya hai, toh Q bahut bada hai, aur voltage E° se kaafi kam hai. Agar dono exactly balanced hain (equilibrium), toh voltage zero hai — aur flow nahi!

RT/nF part sirf ek "unit converter" hai jo concentration ratio ko volts mein badalta hai.

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Connections

• Standard Reduction Potentials — E° values in tables se aati hain
• Gibbs Free Energy and Spontaneity — ΔG = -nFE thermodynamics ko electrochemistry se link karta hai
• Reaction Quotient Q vs Equilibrium Constant K — Q dynamic version hai, K equilibrium par Q hai
• Electrochemical Cells (Galvanic vs Electrolytic) — Nernst galvanic cells par apply hota hai
• Le Chatelier's Principle — concentrations change karna E ko shift karta hai bilkul equilibrium shift karne jaisa
• pH and Half-Cell Potentials — H⁺ concentration Nernst ke zariye E ko affect karta hai
• Battery Discharge Curves — kyun battery run hone par voltage girta hai (Q badhta hai)

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Flashcards

log₁₀ use karke 25°C par Nernst equation kya hai? :: $E=E°-\frac{0.0592}{n}\log Q$

Nernst equation mein Q kya represent karta hai? :: Reaction quotient, product concentrations ka reactant concentrations se ratio (solids/liquids exclude karke).

Concept Map