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

Verified step by step guidance

1. Calculate the change in Gibbs free energy (ΔG) using the formula: ΔG = -RT \ln\left(\frac{C_2}{C_1}\right), where R is the gas constant (8.314 J/mol·K), T is the temperature in Kelvin, C_1 is the initial concentration of K⁺ in the blood plasma, and C_2 is the final concentration of K⁺ in the muscle-cell fluid.
2. Convert the body temperature from Celsius to Kelvin by adding 273.15 to the Celsius temperature. This is necessary because the formula for ΔG requires the temperature to be in Kelvin.
3. Substitute the values into the formula: R = 8.314 J/mol·K, T = 310.15 K (37 °C + 273.15), C_1 = 5.0⨉10^-3 M, and C_2 = 0.15 M.
4. Calculate the natural logarithm (ln) of the ratio of the concentrations (C_2/C_1).
5. The result from the ΔG calculation will give the minimum amount of work that must be used to transfer 1 mol of K⁺ from blood plasma to the cellular fluid, as ΔG also represents the maximum non-expansion work that can be done by a system at constant temperature and pressure.

Key Concepts

Gibbs Free Energy (ΔG)

Gibbs Free Energy (ΔG) is a thermodynamic potential that measures the maximum reversible work obtainable from a thermodynamic system at constant temperature and pressure. It is calculated using the formula ΔG = ΔG° + RT ln(Q), where ΔG° is the standard free energy change, R is the universal gas constant, T is the temperature in Kelvin, and Q is the reaction quotient. A negative ΔG indicates a spontaneous process, while a positive ΔG suggests non-spontaneity.

Concentration Gradient

A concentration gradient refers to the difference in the concentration of a substance across a space or membrane. In this context, the gradient exists between the potassium ion concentrations in blood plasma (5.0×10^-3 M) and muscle-cell fluid (0.15 M). Ions tend to move from areas of higher concentration to areas of lower concentration, and this movement is crucial for understanding the direction and spontaneity of ion transfer across membranes.

Work and Energy Transfer

In thermodynamics, work is defined as the energy transferred when an object is moved by an external force. The minimum amount of work required to transfer ions against their concentration gradient can be calculated using the relationship between Gibbs Free Energy and work. Specifically, the work done on the system is equal to the change in Gibbs Free Energy, which reflects the energy needed to move ions from a region of lower concentration to a region of higher concentration.