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

The work done in transporting ions across a membrane can be calculated using the Gibbs free energy formula: \(\Delta G = \Delta G^0 + RT \ln\frac{Q}{K}\) where: - \(\Delta G\) is the Gibbs free energy change - \(\Delta G^0\) is the standard Gibbs free energy change (0 for this process) - R is the gas constant (8.314 J/mol K) - T is the temperature in Kelvin (in this case, 37°C = 310 K) - Q is the reaction quotient (calculated by dividing the concentration of K+ in muscle cells by the concentration in blood plasma) - K is the equilibrium constant (1 for this process since the concentrations remain constant)

We are given that the concentration of K+ in muscle cells is 0.15 M, and the concentration in blood plasma is 0.0050 M. We can calculate the reaction quotient Q as follows: \(Q = \frac{\text{concentration in muscle cells}}{\text{concentration in blood}} = \frac{0.15 M}{0.0050 M} = 30\)

Using the given formula, we can now find the Gibbs free energy change: \(\Delta G = \Delta G^0 + RT \ln\frac{Q}{K} = 0 + (8.314 J/mol \cdot K)(310 K) \ln(30)\) Solve for \(\Delta G\): \(\Delta G = 8.314 \times 310 \times \ln(30) \approx 2.99 \times 10^4 J/mol\)

The transportation of K+ from blood to cells results in a change in the overall charge balance across the cell membrane. As K+ ions are moved into the cell, more positive charges are transferred inside the cell. In order to maintain charge neutrality, an equivalent amount of anionic charge (negative ions) must be moved across the membrane or an equivalent amount of cationic charge (positive ions) must be moved out of the cell to maintain the overall charge balance. Therefore, the transportation of other ions is necessary to maintain charge neutrality during this process. #Result# The work required to transport 1.0 mol K+ from the blood to the inside of a muscle cell at 37°C is approximately 2.99 x 10^4 J/mol. Yes, the transportation of other ions is necessary to maintain charge neutrality during this process.