Question

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

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

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

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.

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.

Key concepts

Potassium-Ion Concentration

Potassium-ion concentration plays a vital role in maintaining the body's cellular function. The difference in potassium-ion concentrations between the blood plasma and muscle cells creates a gradient essential for various physiological processes.
In most body systems, the blood plasma contains a lower concentration of potassium ions, about \(5.0 \times 10^{-3} \mathrm{M}\). Meanwhile, inside muscle cells, the potassium concentration is much higher at \(0.15 \mathrm{M}\). This steep gradient is crucial for functions like nerve signal transmission and muscle contraction.
Understanding how potassium ions move across membranes helps in exploring how cells maintain their electrical charge and function efficiently. Such knowledge is particularly important in understanding cell metabolism and can assist in diagnosing related health conditions.

Reaction Quotient

The reaction quotient \(Q\) is a helpful concept in chemistry for understanding the ratios of concentrations for certain reactions. It's used to determine the direction in which a reaction will proceed. For the given scenario, the quotient helps us calculate the energy change during the transfer of ions.
For potassium ions, \(Q\) is calculated by dividing the concentration of ions in the muscle cell fluid by that in blood plasma:

• Potassium concentration in muscle cell fluid: \(0.15 \mathrm{M}\)
• Potassium concentration in blood plasma: \(5.0 \times 10^{-3} \mathrm{M}\)

Using these values, \(Q = \frac{0.15}{5.0 \times 10^{-3}}\). This ratio is plugged into the formula for calculating Gibbs free energy \(\Delta G\), which determines the spontaneity of the ion transfer.

Cell Membrane Permeability

Cell membrane permeability refers to the membrane's ability to allow certain substances to pass through while blocking others. This selective permeability is crucial for maintaining cell homeostasis.
In this exercise, the cell membrane is assumed to be permeable only to potassium ions \(\mathrm{K}^{+}\). This means potassium can move freely across the membrane, while other substances might need specialized transport mechanisms or channels.
This characteristic allows cells to maintain specific internal environments different from their external surroundings. By controlling which substances enter and leave, the membrane plays an integral role in regulating cell function, communicating signals, and managing metabolic activities.

Minimum Work Transfer

Minimum work transfer involves calculating the least amount of energy necessary to move ions across the cell membrane against a concentration gradient.
In thermodynamics, the minimum work needed is directly related to the change in Gibbs free energy \(\Delta G\). The absolute value of \(\Delta G\) provides us with the minimum work energy, which ensures processes like ion transport occur efficiently.
As illustrated, the minimum work to transfer one mole of potassium ions from the blood plasma to the muscle cell fluid is approx. \(8560 \text{ J/mol}\). Since \(\Delta G\) was found to be negative, it confirms that such a transfer requires no net input of work, highlighting that it is a spontaneous process. This underscores the body’s efficient use of energy in maintaining necessary physiological gradients.