Many biochemical reactions that occur in cells require relatively high concentrations of potassium ion \(\left(\mathrm{K}^{+}\right)\). The concentration of \(\mathrm{K}^{+}\) in muscle cells is about \(0.15 \mathrm{M}\). The concentration of \(\mathrm{K}^{+}\) in blood plasma is about \(0.0050 \mathrm{M}\). The high internal concentration in cells is maintained by pumping \(\mathrm{K}^{+}\) from the plasma. How much work must be done to transport \(1.0\) mole of \(\mathrm{K}^{+}\) from the blood to the inside of a muscle cell at \(37^{\circ} \mathrm{C}\), normal body temperature? When \(1.0\) mole of \(\mathrm{K}^{+}\) is transferred from blood to the cells, do any other ions have to be transported? Why or why not?

Potassium ion transportation across a membrane can be understood in terms of chemical potential difference, Δμ. The difference in chemical potential acts as the driving force to transport K+ ions from blood to the inside of muscle cells. Under certain conditions, we can treat Δμ as the change in Gibbs free energy (ΔG). To find ΔG, we need to consider the Nernst equation, which shows the relationship between concentration and electrical potential for ions across a membrane: \[ \Delta G = RT \ln \frac{[K^{+}]_{inside}}{[K^{+}]_{outside}} \] Here, R is the universal gas constant (8.314 J/mol K), T is the absolute temperature in Kelvin (37°C + 273.15 = 310.15 K), [K+]inside and [K+]outside are the molar concentrations of potassium ions inside and outside the muscle cells, respectively.

We have all the values to calculate ΔG using the Nernst equation. Plugging them into the formula, we get: \[ \Delta G = (8.314 \,\text{J/mol K})(310.15 \,\text{K}) \ln\left(\frac{0.15\, \text{M}}{0.0050 \,\text{M}}\right) \]

To find the work done, we simply need to multiply ΔG by the number of moles of K+ ions. In this case, it is 1.0 mole: \[ W = (1.0 \, \text{mol}) \times (8.314 \,\text{J/mol K})(310.15 \,\text{K}) \ln\left(\frac{0.15\, \text{M}}{0.0050 \,\text{M}}\right) \approx 18569 \, \text{J} \] So, approximately 18,569 Joules of work must be done to transport 1.0 mole of K+ ions from the blood to the inside of a muscle cell at 37°C.

When 1.0 mole of K+ ions is transferred from blood plasma to the muscle cell, other ions, like sodium ions (Na+), must also be transported to maintain overall electrical neutrality. This happens because living cells use various pumps, such as the sodium-potassium pump (Na+/K+ pump), to maintain the proper ion balance in their intracellular and extracellular environments. The transport of other ions ensures that the net charge across the membrane doesn't change while maintaining the homeostasis of the cell.