pH and conductivity measurement

Why it matters

pH and conductivity are the two most common liquid analysers in industry. pH controls neutralisation of effluent, fermentation, boiler and cooling-water chemistry, food and pharmaceutical processing. Conductivity tracks dissolved salts in boiler feed water, reverse-osmosis permeate and condensate return, and measures acid or alkali concentration in cleaning-in-place systems. Both are electrochemical measurements with characteristic pitfalls — electrode slope, temperature effects, fouling and polarisation — that every instrument engineer meets.

Key ideas

pH.

• pH = −log₁₀ a(H⁺), where a is the hydrogen-ion activity (≈ concentration in mol/L for dilute solutions). At 25 °C neutral water has pH 7.
• Measuring cell: a glass electrode whose thin pH-sensitive glass membrane develops a potential that depends on the H⁺ activity difference across it, and a reference electrode (usually Ag/AgCl in KCl) that supplies a constant potential through a porous liquid junction. Combination electrodes contain both.
• Nernst equation: the cell emf changes by −2.303·R·T/F per pH unit: 59.16 mV/pH at 25 °C, rising in proportion to absolute temperature (66.1 mV/pH at 60 °C). Practical electrodes are made to give about 0 mV at pH 7 (the isopotential point), positive mV in acid and negative mV in alkali.
• High impedance: the glass membrane has a resistance of tens to hundreds of megohms, so the meter needs an input impedance of about 10¹² Ω, short screened cables or a preamplifier at the sensor, and good insulation (dampness ruins readings).
• Calibration: with two (or three) buffers, e.g. pH 7.00 and 4.01 or 10.01. The offset gives the zero (asymmetry) error and the slope gives the electrode efficiency; a slope below about 90–95 % of theoretical indicates an ageing electrode.
• Temperature compensation corrects the Nernst slope only. It does not correct the real change of a solution's pH with temperature.
• Errors: alkaline (sodium) error above pH 12 (reads low), acid error below pH 1, clogged or poisoned reference junction (drift, slow response), coating of the glass, ground loops and streaming potentials in fast or low-conductivity water.

Conductivity.

• Electrolytic conductivity κ (S/m; practical unit µS/cm, with 1 S/m = 10 mS/cm = 10 000 µS/cm) measures total ionic content but not which ions are present. Ultrapure water at 25 °C is about 0.055 µS/cm; drinking water a few hundred µS/cm; sea water about 50 mS/cm.
• Contacting cells: two electrodes of area A spaced L apart; the measured conductance G = 1/R is converted by the cell constant K = L/A (cm⁻¹): κ = G·K. Low-conductivity water needs small K (0.01–0.1 cm⁻¹); concentrated solutions need large K (1–10 cm⁻¹). Four-electrode cells use separate current and voltage electrodes to reduce polarisation and fouling errors.
• AC excitation is essential: DC would polarise the electrodes and electrolyse the solution.
• Inductive (toroidal, electrodeless) sensors: one toroid induces a current loop in the liquid, a second senses it. No electrodes contact the liquid, so they suit concentrated acids, slurries and fouling liquids, but not very low conductivities.
• Temperature: conductivity rises roughly 2 % per °C for most salt solutions (different for acids, alkalis and pure water), so readings are referred to 25 °C.

Formulas

pH = −log₁₀ a(H⁺)

E = E₀ − (2.303·R·T/F)·pH (Nernst; slope S = 2.303·R·T/F)

S(T) = 59.16 mV × (T/298.15 K) (mV per pH)

pH = pH_cal + (E_cal − E)/S (single-point with known slope)

κ = G·K = K/R, K = L/A

κ₂₅ = κ_T / (1 + α·(T − 25)) (linear temperature compensation)

ρ_e = 1/κ (resistivity)

Symbols: a(H⁺) = hydrogen-ion activity (mol/L); E = cell emf (V or mV); E₀ = standard potential (V); R = 8.314 J/(mol·K); T = absolute temperature (K) in the Nernst slope; F = 96 485 C/mol; S = slope (mV/pH); κ = conductivity (S/m or µS/cm); G = conductance (S); R (in κ = K/R) = measured resistance (Ω); K = cell constant (cm⁻¹ or m⁻¹); L = electrode spacing; A = electrode area; α = temperature coefficient (about 0.02 /°C for many salts; take from data for the solution); T in the compensation formula in °C; ρ_e = resistivity (Ω·m).

Worked examples

Example 1 (standard): two-buffer pH calibration. Given: at 25 °C the electrode reads −3.0 mV in pH 7.00 buffer and +170.0 mV in pH 4.01 buffer. A sample reads +80.0 mV.

1. Slope = (170.0 − (−3.0))/(4.01 − 7.00) = 173.0/(−2.99) = −57.86 mV/pH.
2. Efficiency = 57.86/59.16 = 97.8 % (acceptable).
3. pH = 7.00 + (E − E₇)/slope = 7.00 + (80.0 − (−3.0))/(−57.86) = 7.00 − 1.434.
4. Answer: pH ≈ 5.57, with an offset of −3 mV and a slope of 97.8 %.

Example 2 (GATE level): temperature effect on the Nernst slope. Given: an ideal electrode (0 mV at pH 7) in a process at 60 °C reads −198 mV. Find the pH, and the error if the meter wrongly used the 25 °C slope.

1. S(60 °C) = 59.16 × (333.15/298.15) = 59.16 × 1.1174 = 66.10 mV/pH.
2. pH = 7 + 198/66.10 = 7 + 2.995 = 9.995 ≈ 10.0.
3. Using 59.16 mV/pH: pH = 7 + 198/59.16 = 7 + 3.347 = 10.35.
4. Answer: true pH ≈ 10.0; without temperature compensation the meter reads ≈ 10.35 (error +0.35 pH). The error grows with distance from pH 7 because the zero is at pH 7.

Example 3 (conductivity with compensation). Given: cell constant K = 0.1 cm⁻¹; measured resistance R = 2000 Ω at 35 °C; α = 0.02 /°C.

1. κ₃₅ = K/R = 0.1/2000 = 5.0 × 10⁻⁵ S/cm = 50 µS/cm.
2. κ₂₅ = 50/(1 + 0.02 × (35 − 25)) = 50/1.20 = 41.7 µS/cm.
3. Answer: κ₂₅ ≈ 41.7 µS/cm.

Common mistakes

• Treating the pH electrode slope as fixed at 59 mV/pH regardless of temperature.
• Thinking automatic temperature compensation gives the solution's pH at 25 °C; it only corrects the electrode slope.
• Connecting a pH electrode to an ordinary voltmeter (input impedance too low) or using long unscreened cable.
• Using DC to measure conductivity.
• Mixing S/m, mS/cm and µS/cm (factors of 10 and 10 000).
• Choosing a cell constant unsuited to the conductivity range, or a contacting cell in a fouling, concentrated stream where a toroidal sensor belongs.

For GATE IN

Numericals: Nernst slope at a given temperature, pH from electrode emf, two-buffer calibration, conductivity from cell constant and resistance, and temperature-compensated conductivity. Conceptual questions: why the glass electrode needs a high-impedance amplifier, the role of the reference electrode and junction, AC excitation, and contacting versus inductive conductivity sensors. Practise logarithms and unit conversions carefully.

Quick check

1. What is the Nernst slope at 25 °C?
2. [H⁺] = 10⁻³ mol/L. What is the pH?
3. A cell with K = 1 cm⁻¹ measures G = 500 µS. What is κ?
4. Why must the pH meter have a very high input impedance?
5. Convert 2 S/m to µS/cm.

Answers: 1. 59.16 mV/pH. 2. 3. 3. 500 µS/cm. 4. The glass membrane has a resistance of tens to hundreds of megohms; any current drawn would load the cell and lower the reading. 5. 20 000 µS/cm.