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Chapter 19: Problem 102

The potassium-ion concentration in blood plasma is about $5.0 \times 10^{-3} \mathrm{M}$, 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 \(\mathrm{K}^{+} ?\)

Short Answer

Expert verified
In short, the change in Gibbs free energy (ΔG) for the transfer of 1 mol of potassium ion (K+) from blood plasma to muscle cell fluid is approximately -8560 J/mol, indicating a spontaneous process. The minimum amount of work required to transfer this K+ is about 8560 J/mol.

Step by step solution

01

Calculate the reaction quotient Q

We are given the concentration of potassium ions in the blood plasma as 5.0 × 10^(-3) M and in the muscle cell fluid as 0.15 M. Since K+ is the only species being transferred, the reaction quotient Q can be written as: Q = [K+ in muscle cell fluid] / [K+ in blood plasma] Insert the values: Q = 0.15 M / 5.0 × 10^(-3) M
02

Calculate ΔG

We will use the formula described in the analysis: ΔG = ΔGº + RT * ln(Q) Insert ΔGº (0) and the calculated value of Q: ΔG = 0 + (8.314 J/(mol·K)) * 310 K * ln(0.15 M / 5.0 × 10^(-3) M) Calculate ΔG, which will give the change in Gibbs free energy for the transfer of K+ ions: ΔG ≈ -8560 J/mol This negative value indicates that the transfer of potassium ions from blood plasma to muscle cell fluid is spontaneous.
03

Find the minimum work needed

The minimum work required to transfer the K+ ion can be determined from the calculated ΔG. The minimum work is equal to the absolute value of ΔG: Minimum Work = |ΔG| ≈ 8560 J/mol The minimum amount of work required to transfer 1 mol of K+ ions from blood plasma to the muscle cell fluid is approximately 8560 J/mol.

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Most popular questions from this chapter

Predict the sign of the entropy change of the system for each of the following reactions: (a) $\mathrm{CO}(g)+\mathrm{H}_{2}(g) \longrightarrow C(s)+\mathrm{H}_{2} \mathrm{O}(g)$ (b) $2 \mathrm{O}_{2}(g)+\mathrm{N}_{2}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)$ (c) $\mathrm{NH}_{4} \mathrm{Cl}(s) \longrightarrow \mathrm{HCl}(g)+\mathrm{NH}_{3}(g)$ (d) $2 \mathrm{C}_{2} \mathrm{H}_{4}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{C}_{2} \mathrm{H}_{4} \mathrm{O}(g)$

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(a) For each of the following reactions, predict the sign of \(\Delta H^{*}\) and \(\Delta S^{\circ}\) without doing any calculations. (b) Based on your general chemical knowledge, predict which of these reactions will have \(K>1\) at \(25^{\circ} \mathrm{C} .(\mathbf{c})\) In each case, indicate whether \(K\) should increase or decrease with increasing temperature. (i) \(2 \mathrm{Fe}(s)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{FeO}(s)\) (ii) \(\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{Cl}(g)\) (iii) $\mathrm{NH}_{4} \mathrm{Cl}(s) \rightleftharpoons \mathrm{NH}_{3}(g)+\mathrm{HCl}(g)$ (iv) $\mathrm{CO}_{2}(\mathrm{~g})+\mathrm{CaO}(s) \rightleftharpoons \mathrm{CaCO}_{3}(s)$

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