Cell EMF & Internal Resistance lab report

1. Title and aim

Internal Resistance and Electromotive Force of a Cell: Terminal Voltage and Load Characteristics. To determine the electromotive force (EMF) and internal resistance of an electrochemical primary cell by measuring terminal potential difference as a function of load current, evaluate the short-circuit current and maximum power transfer, and investigate instrument loading effects.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of internal resistance of 2.500 Ω, differing by 0.000% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• Primary Electrochemical Cell: Provides a stable chemical electromotive force (nominal open-circuit EMF ℰ = 1.500 V) with an inherent internal electrolytic and polarization resistance (r = 2.500 Ω) arising from finite ionic mobility in the ammonium chloride / zinc electrolyte paste.
• Precision Decade Resistance Box: Provides calibrated non-inductive standard load resistances from 2.0 Ω to 50.0 Ω (tolerance ±0.1%), allowing smooth and repeatable adjustment of the external circuit current without introducing reactive impedance.
• High-Impedance Digital DC Voltmeter: Measures the terminal potential difference V across the cell output posts with 0.001 V resolution and an input impedance exceeding 10 MΩ to ensure that voltmeter loading current is strictly negligible (< 0.15 μA).
• Low-Resistance Precision DC Ammeter: Measures the closed-loop load current I from 0.0 mA to 350.0 mA with 0.1 mA resolution, engineered with a low-burden shunt resistance (R_A ≤ 0.05 Ω) to minimize insertion voltage drops.
• Single-Pole Low-Resistance Knife Switch: Closes and opens the discharge circuit intermittently with negligible contact resistance (< 5 mΩ), preventing prolonged cell discharge and minimizing thermal heating or cumulative polarization during observation intervals.
• Heavy-Gauge Copper Connecting Leads: Forms robust low-resistance electrical interconnections between circuit components with total lead resistance under 0.02 Ω, ensuring that all measured potential drops represent the cell and intended load.
• Open-circuit electromotive force ℰ = 1.5000 V (calibrated against Weston standard cadmium cell)
• Reference internal resistance r_ref = 2.500 Ω (at 20.0 °C ambient room temperature)
• Load current range sampled: I = 50.0 mA to 300.0 mA in 50.0 mA increments
• Voltmeter input impedance Z_in ≥ 10 MΩ (negligible loading correction < 0.01%)
• Ammeter internal shunt burden resistance R_A ≤ 0.050 Ω
• Intermittent discharge protocol: 5-second measurement pulses separated by 60-second recovery intervals

4. Theory and graph choice

The electromotive force (ℰ) of an electrical source is the total energy converted per unit charge in driving electricity around a closed circuit. Because physical charge carriers must traverse internal electrolytic solutions, active paste masses, and electrode barriers, every practical source possesses an inherent internal resistance (r). Consequently, when current (I) is drawn by an external load, an internal ohmic potential drop (I · r) is consumed inside the source itself, causing the terminal potential difference (V) to fall below ℰ according to V = ℰ − I · r. By measuring terminal voltage across a series of calibrated load currents and applying linear regression, students determine both the internal resistance r from the negative slope and the open-circuit EMF ℰ from the vertical intercept, verifying the fundamental governing relations of electric currents.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units Ω (V A⁻¹).

5. Method

1. Inspect the open-circuit cell using the high-impedance digital voltmeter with the main knife switch open; record the quiescent open-circuit terminal voltage to verify baseline electromotive force ℰ₀ = 1.500 V prior to drawing discharge current.
2. Connect the circuit in series: primary cell positive terminal to ammeter input, ammeter output to the decade resistance box, and resistance box back to the cell negative terminal through the knife switch. Connect the high-impedance voltmeter directly across the cell battery posts (Kelvin voltage sensing configuration) to exclude lead resistance drops.
3. Set the decade resistance box to its highest load setting (producing minimum initial current I = 50.0 mA). Momentarily close the knife switch for 3 to 5 seconds, read both the ammeter current I and the digital voltmeter terminal potential difference V, and immediately open the switch to prevent chemical exhaustion or polarization buildup.
4. Allow the cell a 60-second recovery period for ionic diffusion to re-establish chemical equilibrium at the electrode interfaces. Step down the load resistance progressively to draw currents of 100.0 mA, 150.0 mA, 200.0 mA, 250.0 mA, and 300.0 mA, recording the stabilized terminal voltage V at each setting.
5. Tabulate the paired values of load current I (in amperes) and terminal potential difference V (in volts). Plot terminal voltage V on the vertical axis against current I on the horizontal axis; fit an ordinary least-squares straight line to determine the slope m and the vertical intercept b.
6. Extract the internal resistance as r = −m and the true electromotive force as ℰ = b. Calculate the extrapolated short-circuit current I_sc = ℰ / r, evaluate the maximum deliverable external power P_max = ℰ² / (4r), and compare the fitted parameters against manufacturer reference specifications.

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0000%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 0.00000; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 0.00000. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 0.00000.

Propagating only the relevant fit coefficient gives a fit-only standard error of 0.00000 Ω. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Transient concentration polarization and hydrogen bubble overpotential at the cathode during current flow (minimized by restricting discharge duration to 3-second pulses followed by 60-second chemical relaxation intervals), digital voltmeter finite input impedance loading (negligible < 0.01% with 10 MΩ instrument vs > 0.5% with 20 kΩ analog meters), ammeter internal shunt burden resistance (R_A ≈ 0.050 Ω adding directly into the series loop), knife-switch contact resistance fluctuations (±5 mΩ), connecting lead resistance (eliminated by connecting voltmeter sensing leads directly to cell terminal lugs via Kelvin 4-wire geometry), and electrolyte temperature drift (+0.03 Ω/°C negative temperature coefficient of resistance) govern the uncertainty envelope. Ordinary least-squares regression of terminal voltage V against discharge current I averages out meter digit quantization and contact variations, with the slope isolating the true internal resistance r and the vertical intercept isolating the open-circuit electromotive force ℰ.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Close the discharge knife switch only for brief 3 to 5 second intervals to avoid cumulative concentration polarization and internal Joule heating.
• Allow at least 60 seconds between successive measurement pulses for chemical diffusion to restore baseline equilibrium at electrode interfaces.
• Connect voltmeter leads directly to the battery posts (Kelvin sensing) rather than to ammeter terminals to prevent lead resistance errors.
• Never place an ammeter directly across the cell without an external load resistor; short-circuit currents can damage meters and rupture cells.
• Ensure all terminal thumb-screws and binding posts are firmly tightened to eliminate spurious contact resistance fluctuations.

Model calculations assume constant ambient temperature (20.0 °C), negligible battery heating, and ideal linear Butler-Volmer kinetics without irreversible chemical degradation or deep discharge hysteresis. Real secondary and primary cells exhibit transient polarization relaxation times (τ ~ 5 to 30 s), non-linear Butler-Volmer overpotentials at very high discharge rates, and finite internal thermal capacitance.

12. Conclusion

The worked example illustrates how to obtain internal resistance from measurements, graph variables and a fitted coefficient. The estimate is 2.500 Ω. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.