The potassium-ion concentration in blood plasma is about 5.0 ×10^-3 M, whereas the concentration

step 1 1 .02 in chapter 19. So the problem tells you that the concentration of potassium ion in blood p

The potassium-ion concentration in blood plasma is about 5.0...

The potassium-ion concentration in blood plasma is about $5.0 \times 10^{-3} M,$ whereas the concentration in muscle-cell fluid is much greater $(0.15 \mathrm{M})$. The plasma and intracellular fluid are separated by the cell membrane, which we assume is permeable only to $\mathrm{K}^{+}$. (a) What is $\Delta G$ for the transfer of $1 \mathrm{~mol}$ of $\mathrm{K}^{+}$ from blood plasma to the cellular fluid at body temperature $37^{\circ} \mathrm{C} ?$ (b) What is the minimum amount of work that must be used to transfer this $\mathrm{K}^{+} ?$

The potassium-ion concentration in blood plasma is about $5.0 \times 10^{-3} M,$ whereas the concentration in muscle-cell fluid is much greater $(0.15 \mathrm{M})$. The plasma and intracellular fluid are separated by the cell membrane, which we assume is permeable only to $\mathrm{K}^{+}$.
(a) What is $\Delta G$ for the transfer of $1 \mathrm{~mol}$ of $\mathrm{K}^{+}$ from blood plasma to the cellular fluid at body temperature $37^{\circ} \mathrm{C} ?$ (b) What is the minimum amount of work that must be used to transfer this $\mathrm{K}^{+} ?$

Key Concepts

Reversible Work and Minimum Energy Requirement

The concept of reversible work is tied to the idea of the minimum work required to drive a process. In a reversible (idealized) process, the work done equals the change in free energy. This establishes a fundamental limit on the minimum energy input necessary for processes like ion transport against a concentration gradient, ensuring that the calculated work is the smallest possible amount needed.

Membrane Permeability

Membrane permeability refers to the selective ability of a cell membrane to allow certain molecules or ions to pass while restricting others. In problems involving ion transfer, the assumption that the membrane is permeable only to a particular ion simplifies the analysis by ensuring that only that ion contributes to the change in free energy, thus eliminating complications from other ionic species.

Chemical Potential and Concentration Gradients

Chemical potential represents the energy per mole associated with a substance and varies with concentration. A concentration gradient, where there is a difference in chemical potential between two regions, is the driving force for the spontaneous movement of ions. The mathematical form of the free energy change, involving the natural logarithm of the concentration ratio, directly reflects the influence of the gradient on the system’s energy.

Gibbs Free Energy

Gibbs free energy is a thermodynamic potential that quantifies the maximum work obtainable from a system at constant temperature and pressure. In the context of ionic transfer across a membrane, it is used to determine the energetic favorability or cost of moving an ion from one side of the membrane to the other, using the formula ?G = RT ln([final concentration]/[initial concentration]).

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