Write the Nernst equation for the cell reaction in the Daniel cell. How will the ECell be affected when concen

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E_{Cell}

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Concept understanding — Cell Representation Nernst Equation

Cell Representation and the Nernst Equation: From Intuition to Precision

Imagine you have a Daniell cell — a zinc rod in zinc sulphate solution connected by a salt bridge to a copper rod in copper sulphate solution. You know it produces a voltage. But what happens if you dilute the copper sulphate solution? Or if you change the temperature? The voltage changes. The Nernst equation is the tool that tells you exactly how much it changes.

The Intuition First

A battery works because the two half-cells "want" to react — zinc wants to lose electrons, copper ions want to gain them. This "want" is measured as a tendency, or potential. But the strength of that tendency depends on how crowded the ions are.

Think of it like this: If you have a room full of people who all want to leave (like zinc ions wanting to form), the push to get out is stronger when the room is packed. If the room is nearly empty, the push is weaker. Similarly, for copper ions wanting to enter the metal (gain electrons), the pull is stronger when there are many copper ions around, and weaker when there are few.

The Nernst equation quantifies this: the actual cell potential depends on the concentrations (or activities) of the ions involved.

The Precise Statement

For a general cell reaction:

aA + bB \rightarrow cC + dD

The cell potential $E$ under non-standard conditions is given by:

E = E^\circ - \frac{RT}{nF} \ln Q

Where:

$E$ = cell potential under the given conditions (in volts)
$E^\circ$ = standard cell potential (when all reactants/products are at 1 M, 1 atm, 25°C)
$R$ = universal gas constant (8.314 J/mol·K)
$T$ = temperature in Kelvin
$n$ = number of moles of electrons transferred in the balanced half-reactions
$F$ = Faraday constant (96,485 C/mol)
$Q$ = reaction quotient = $\frac{[C]^c [D]^d}{[A]^a [B]^b}$ (using concentrations for dilute solutions)

At 25°C (298 K), the equation simplifies to a very practical form:

E = E^\circ - \frac{0.0591}{n} \log_{10} Q

The 0.0591 comes from $\frac{2.303RT}{F}$ at 298 K. Notice it uses $\log_{10}$ (common log), not natural log.

Cell Representation: How We Write It

In electrochemistry, we represent a cell with a shorthand notation. For the Daniell cell:

\text{Zn(s)} \mid \text{Zn}^{2+}(\text{aq}) \parallel \text{Cu}^{2+}(\text{aq}) \mid \text{Cu(s)}

The single vertical line $\mid$ represents a phase boundary (solid electrode | solution). The double line $\parallel$ represents the salt bridge.

The anode (oxidation) is written on the left, the cathode (reduction) on the right. Electrons flow from left to right in the external circuit.

Applying the Nernst Equation to a Cell Representation

For the Daniell cell, the half-reactions are:

Anode (oxidation): $\text{Zn(s)} \rightarrow \text{Zn}^{2+}(\text{aq}) + 2e^-$
Cathode (reduction): $\text{Cu}^{2+}(\text{aq}) + 2e^- \rightarrow \text{Cu(s)}$

Overall: $\text{Zn(s)} + \text{Cu}^{2+}(\text{aq}) \rightarrow \text{Zn}^{2+}(\text{aq}) + \text{Cu(s)}$

Here $n = 2$ (two electrons transferred). The reaction quotient is:

Q = \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]}

So the Nernst equation becomes:

E = E^\circ - \frac{0.0591}{2} \log_{10} \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]}

Note

Solids (Zn, Cu) do not appear in $Q$ because their concentrations are constant (activity = 1).

A Worked Example

Suppose you have a Daniell cell where $[\text{Zn}^{2+}] = 0.1$ M and $[\text{Cu}^{2+}] = 1.0$ M at 25°C. $E^\circ$ for the cell is 1.10 V.

$E = 1.10 - \frac{0.0591}{2} \log_{10} \frac{0.1}{1.0}$ …

Why this formula?

Cell Representation & the Nernst Equation: Why It Works

The Core Question

Why does a cell's voltage change when concentrations change? The Nernst equation answers this — but the reason lies in the link between chemical free energy and electrical work.

1. The Fundamental Link: Gibbs Free Energy & Cell Potential

A galvanic cell does electrical work. The maximum useful work a cell can do equals the change in Gibbs free energy ( $\Delta G$ ):

\Delta G = -nFE_{\text{cell}}

Where:

$n$ = moles of electrons transferred
$F$ = Faraday constant ( $96,485 \, \text{C mol}^{-1}$ )
$E_{\text{cell}}$ = cell potential (volts)

Why negative? A spontaneous reaction has $\Delta G < 0$ and $E_{\text{cell}} > 0$ — the negative sign makes this consistent.

2. The Chemical Side: $\Delta G$ Depends on Concentration

For a general redox reaction:

aA + bB \rightarrow cC + dD

The Gibbs free energy under non-standard conditions is:

\Delta G = \Delta G^\circ + RT \ln Q

Where $Q$ is the reaction quotient:

Q = \frac{[C]^c [D]^d}{[A]^a [B]^b}

Why this form? It comes from the relationship between chemical potential and concentration — the entropy of mixing drives concentration dependence.

3. Combining Both Sides: The Derivation

Set the electrical work equal to the chemical free energy change:

-nFE_{\text{cell}} = -nFE^\circ_{\text{cell}} + RT \ln Q

Divide both sides by $-nF$ :

E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF} \ln Q

This is the Nernst equation.

4. The "Why" in Plain Terms

Concept	Physical Meaning
$E^\circ_{\text{cell}}$	Voltage when all species are at 1 M (standard state)
$-\frac{RT}{nF} \ln Q$	Correction factor — adjusts voltage for real concentrations
$Q$	Tells you how far the reaction is from equilibrium

Key insight: When $Q = K$ (equilibrium), $E_{\text{cell}} = 0$ — the battery is dead because no net reaction occurs.

5. The Common Form (log base 10)

At $25^\circ \text{C}$ ( $298 \, \text{K}$ ):

\frac{RT}{F} \ln 10 \approx 0.0592 \, \text{V}

So:

E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{0.0592}{n} \log_{10} Q

Why convert to log? Exam convenience — most concentration values are powers of 10.

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The Nernst equation for the Daniel cell is $E_{cell} = E^\circ_{cell} - \frac{0.059}{2} \log \frac{[Zn^{2+}]}{[Cu^{2+}]}$ at 298 K. Increasing $[Zn^{2+}]$ increases the log term, which decreases $E_{cell}$ .

The Daniel cell is the classic example of a galvanic cell — zinc and copper electrodes in their respective sulfate solutions, connected by a salt bridge. The cell reaction is:

Zn(s) + Cu^{2+}(aq) \rightarrow Zn^{2+}(aq) + Cu(s)

The driving force for this reaction is the difference in reduction potentials. Zinc is more reactive (easier to oxidise), so it acts as the anode. Copper ions are more easily reduced, so copper is the cathode.

The Nernst equation lets us calculate the actual cell potential under non-standard conditions — when concentrations aren't 1 M. It adjusts the standard potential $E^\circ_{cell}$ by a term that depends on the reaction quotient $Q$ .

aA + bB \rightarrow cC + dD

Now let's apply this to the Daniel cell step by step.

Identify $n$ , the number of electrons transferred.

In the balanced reaction $Zn + Cu^{2+} \rightarrow Zn^{2+} + Cu$ , each zinc atom loses 2 electrons, and each copper ion gains 2 electrons. So $n = 2$ .
Write the reaction quotient $Q$ .

For the reaction as written, $Q = \frac{[Zn^{2+}]}{[Cu^{2+}]}$ . Solids (Zn and Cu) have activity = 1, so they don't appear.
Plug into the Nernst equation.

At 298 K:

E_{cell} = E^\circ_{cell} - \frac{0.059}{2} \log \frac{[Zn^{2+}]}{[Cu^{2+}]}

This is the required Nernst equation for the Daniel cell.

Now analyse the effect of increasing $[Zn^{2+}]$ .

Look at the log term: $\log \frac{[Zn^{2+}]}{[Cu^{2+}]}$ . If $[Zn^{2+}]$ increases while $[Cu^{2+}]$ stays the same, the fraction becomes larger, so $\log$ becomes larger (less negative, or more positive).

Since this log term is subtracted from $E^\circ_{cell}$ , a larger log term means a smaller $E_{cell}$ . …

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Common Mistakes: Cell Representation & Nernst Equation (Daniel Cell)

Mistake 1: Writing the Nernst equation with wrong sign or form

The error:

Students often write:

E_{cell} = E^\circ_{cell} - \frac{0.0591}{n} \log \frac{[\text{Anode}]}{[\text{Cathode}]}

...but then flip the ratio or forget the sign.

Why it happens:

They memorise the formula without understanding that the reaction quotient $Q$ is always products over reactants — and in a Daniel cell, the spontaneous reaction is:

\text{Zn(s)} + \text{Cu}^{2+}(aq) \rightarrow \text{Zn}^{2+}(aq) + \text{Cu(s)}

So $Q = \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]}$ .

How to avoid:

Always write the balanced cell reaction first, then:

E_{cell} = E^\circ_{cell} - \frac{0.0591}{n} \log Q

where $Q = \frac{[\text{products}]}{[\text{reactants}]}$ (excluding solids). For Daniel cell:

E_{cell} = E^\circ_{cell} - \frac{0.0591}{2} \log \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]}

Mistake 2: Confusing anode and cathode in the ratio

The error:

Writing $\log \frac{[\text{Cu}^{2+}]}{[\text{Zn}^{2+}]}$ instead of the correct $\log \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]}$ .

Why it happens:

Students think "cathode over anode" because reduction happens at cathode — but the Nernst equation uses the reaction quotient, not electrode labels.

How to avoid:

Remember: Zn is oxidised (anode, loses electrons) → Zn²⁺ appears in products. Cu²⁺ is reduced (cathode) → appears in reactants. So:

Products: Zn²⁺ (numerator)
Reactants: Cu²⁺ (denominator)

Mistake 3: Forgetting $n = 2$ in the Daniel cell

The error:

Using $n = 1$ or writing $0.0591$ without dividing by $n$ .

Why it happens:

Rushing — the Daniel cell involves transfer of 2 electrons:

\text{Zn} \rightarrow \text{Zn}^{2+} + 2e^-

\text{Cu}^{2+} + 2e^- \rightarrow \text{Cu}

How to avoid:

Always count electrons from the balanced half-reactions. For Daniel cell, $n = 2$ always.

Mistake 4: Misinterpreting the effect of increasing [Zn²⁺]

The error:

Saying "E_cell increases" or "no change" when [Zn²⁺] is increased.

Why it happens:

Students think "more ions = more reaction" without checking the Nernst equation.

The correct reasoning:

From the Nernst equation:

E_{cell} = E^\circ_{cell} - \frac{0.0591}{2} \log \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]}

If [Zn²⁺] increases:

The fraction $\frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]}$ increases
$\log$ of a larger number is more positive
Subtracting a larger positive number → $E_{cell}$ decreases

How to avoid:

Always plug into the equation qualitatively:

More products → larger $Q$ → more negative $\log$ term → lower $E_{cell}$

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Q.Write the Nernst equation for the cell reaction in the Daniel cell. How will the $E_{Cell}$ be affected when concentration of $Zn^{2+}$ ions is increased?

Concept understanding — Cell Representation Nernst Equation

Cell Representation and the Nernst Equation: From Intuition to Precision

The Intuition First

The Precise Statement

Cell Representation: How We Write It

Applying the Nernst Equation to a Cell Representation

A Worked Example

Why this formula?

Cell Representation & the Nernst Equation: Why It Works

The Core Question

1. The Fundamental Link: Gibbs Free Energy & Cell Potential

2. The Chemical Side: $\Delta G$ Depends on Concentration

3. Combining Both Sides: The Derivation

4. The "Why" in Plain Terms

5. The Common Form (log base 10)

Method: Nernst Equation for Cell Potential

Steps:

Common Mistakes: Cell Representation & Nernst Equation (Daniel Cell)

Mistake 1: Writing the Nernst equation with wrong sign or form

Mistake 2: Confusing anode and cathode in the ratio

Mistake 3: Forgetting $n = 2$ in the Daniel cell

Mistake 4: Misinterpreting the effect of increasing [Zn²⁺]

Q.Write the Nernst equation for the cell reaction in the Daniel cell. How will the ECellE_{Cell}ECell​ be affected when concentration of Zn2+Zn^{2+}Zn2+ ions is increased?

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Q.Write the Nernst equation for the cell reaction in the Daniel cell. How will the $E_{Cell}$ be affected when concentration of $Zn^{2+}$ ions is increased?

Unlock everything free for 14 days