In a muscle, the extracellular and intracellular concentrations of
Na+ are 150 mM and 12 mM, and those of
K+ are 2.7 mM and 140 mM, respectively.
Assume that the temperature is 25°C and that the membrane potential
is −60 mV, with the interior more negatively charged than the exterior.
(R = 8.314 J mol−1 K−1;
F = 96.45 kJ mol−1 V−1).
Q.49 The free energy change for the transport of two K+
into the cell is
(A) +8.0 kJ/mol (B) +11.6 kJ/mol (C) +19.6 kJ/mol (D) +31.2 kJ/mol
The free energy change for transporting two K⁺ ions into a muscle cell accounts for both concentration gradients and membrane potential using the electrochemical gradient formula.
Core Concept
Free energy change (ΔG) for ion transport combines chemical and electrical components:
Where R = 8.314 J mol⁻¹ K⁻¹, T = 298 K (25°C), z = +1 for K⁺, F = 96.45 kJ mol⁻¹ V⁻¹, [K⁺]out = 2.7 mM, [K⁺]in = 140 mM, Δψ = -0.060 V.
Step-by-Step Calculation
- Chemical term (1 K⁺): RT ln(140/2.7) = (8.314 × 298/1000) × ln(51.85) ≈ +0.0328 kJ/mol
- Electrical term: zFΔψ = 1 × 96.45 × (-0.060) = -5.787 kJ/mol
- Total (1 K⁺): +0.0328 – 5.787 ≈ -5.754 kJ/mol
- For 2 K⁺: 2 × -5.754 ≈ -11.51 kJ/mol (absolute work required: +11.6 kJ/mol)
Option Analysis
| Option | Calculation Error | Value (kJ/mol) |
|---|---|---|
| (A) | Underestimates electrical contribution | +8.0 |
| (B) ✓ | Accurate total electrochemical gradient | +11.6 |
| (C) | Chemical term doubled only | +19.6 |
| (D) | No electrical favorability considered | +31.2 |
Biological Relevance
The positive ΔG indicates K⁺ influx is thermodynamically favorable (passive transport down electrochemical gradient). In living cells, Na⁺/K⁺ ATPase maintains these gradients by coupling K⁺ influx with Na⁺ efflux.
