Question

The distribution of \(\mathrm{Na}^{+}\)ions across a typical biological membrane is \(10 . \mathrm{mmol} \cdot \mathrm{L}^{-1}\) inside the cell and \(140 \mathrm{mmol} \cdot \mathrm{L}^{-1}\) outside the cell. At equilibrium the concentrations would be equal, but in a living cell the ions are not at equilibrium. What is the Gibbs free energy difference for \(\mathrm{Na}^{+}\)ions across the membrane at \(37^{\circ} \mathrm{C}\) (normal body temperature)? The concentration differential must be maintained by coupling to reactions that have at least that difference of Gibbs free energy.

Short answer

To find the Gibbs free energy difference, first convert Celsius to Kelvin, calculate the membrane potential using the Nernst equation, then apply that potential to the Gibbs free energy equation. The final answer is the product of the negative of Faraday's constant, the number of ions (assuming 1 mol), and the calculated membrane potential.

Step-by-step solution

Understanding Gibbs Free Energy Change

To calculate the Gibbs free energy change for the movement of ions across a membrane, we use the Nernst equation: \(\triangle G = -nFE\), where \(n\) is the number of moles of ions, \(F\) is Faraday's constant (96485 C/mol), and \(E\) is the membrane potential in volts. The membrane potential \(E\) can be calculated using the equation: \(E = (RT/zF) \ln(C_{out}/C_{in})\), where \(R\) is the gas constant (8.314 J/mol K), \(T\) is the temperature in Kelvin, \(z\) is the charge of the ion (for Na\textsuperscript{+}, it is +1), \(C_{out}\) and \(C_{in}\) are the concentrations of the ion outside and inside the cell, respectively. As we are given concentrations in mmol, we will need to convert them to mol in order to use the equation correctly.

Conversion of Temperature to Kelvin

First, we need to convert the temperature from Celsius to Kelvin since the Nernst equation requires the temperature in Kelvin: \(T(K) = T(\degree C) + 273.15\). Hence, for the body temperature of \(37\degree C\), the temperature in Kelvin will be \(37 + 273.15 = 310.15 K\).

Calculate the Membrane Potential (E)

Now we will calculate the membrane potential using the given concentrations and the formula \(E = (RT/zF) \ln(C_{out}/C_{in})\). Substituting the given values, we get \(E = (8.314 \cdot 310.15)/(1 \cdot 96485) \ln(140/10)\). The natural logarithm of \(140/10 = 14\) can be calculated as ln(14).

Calculate Gibbs Free Energy Change

After finding \(E\), we can use it to calculate the Gibbs free energy change using \(\triangle G = -nFE\). Since the question is concerned with an amount of \(1 mol\) ions (which is implicit due to the usage of molar concentrations), \(n\) can be considered as 1. We use the Faraday's constant \(F = 96485 C/mol\), so, \(\triangle G = -(1)(96485)(E)\).

Calculate and Present Final Answer

By calculating \(E\) from the previous steps and then substituting the value into the Gibbs free energy equation, we obtain the Gibbs free energy difference for \(\text{Na}^{+}\) ions across the membrane.

Key concepts

Understanding the Nernst Equation

The Nernst equation is a fundamental relationship used in biochemistry and physiology to determine the membrane potential based on ion concentration gradients. Vital to our understanding of how cells maintain their electrical properties, the Nernst equation provides insight into the behavior of ions and their distribution across biological membranes.

The equation itself can be given as: \[E = \frac{{RT}}{{zF}} \ln\left(\frac{{C_{{out}}}}{{C_{{in}}}}\right)\]where:

• \(E\) represents the membrane potential,
• \(R\) is the universal gas constant (8.314 J/mol K),
• \(T\) is the absolute temperature in Kelvin,
• \(z\) is the valence or charge number of the ion,
• \(F\) is Faraday's constant (96485 C/mol),
• \(C_{{out}}\) and \(C_{{in}}\) are the ion concentrations outside and inside the cell, respectively.

Applying the Nernst equation allows us to calculate the intrinsic electric potential that would be generated by a specific ion gradient, assuming other ions do not contribute to the membrane potential. Understanding this equation is critical when investigating how cells harness energy from ionic differences to perform complex functions.

Calculating Membrane Potential

Membrane potential refers to the voltage difference across a cell's membrane, primarily due to the distribution of ions. It is an essential feature in the functioning of neurons, muscle cells, and electrolyte balance in all cells. Membrane potential can be calculated using the Nernst equation, as demonstrated in the example.

To determine the membrane potential, it's important to consider both the concentration of ions on either side of the membrane and the temperature of the system. After converting the temperature to Kelvins and inserting the ion concentrations and other constants into the Nernst equation, the membrane potential \(E\) can be calculated, revealing the electrochemical driving force for the ion's movement across the membrane.

Practical Application

The actual calculation requires a keen appreciation of logarithmic functions since we deal with ratios of concentrations. For instance:

\(E = \frac{{(8.314 \cdot T)}}{{(z \cdot 96485)}} \ln\left(\frac{{C_{{out}}}}{{C_{{in}}}}\right)\)
Through this simplified equation, we can appreciate the quantitative relationship between ion gradients and membrane potentials, which serves as a cornerstone in topics such as neurophysiology and pharmacology.

The Role of Ion Concentration Gradients

Ion concentration gradients are differences in the concentration of ions across a biological membrane. These gradients are critical for various cell functions, including generating membrane potentials, transmitting nerve impulses, and muscle contraction.

Gradients arise from the actions of transport proteins and ion channels embedded in the cell membrane which actively or passively control the movement of ions such as \(\text{Na}^{+}\), \(\text{K}^{+}\), \(\text{Ca}^{2+}\), and \(\text{Cl}^{-}\). An interesting point to note is that these gradients require energy to be maintained, often derived from ATP hydrolysis or coupled reactions, emphasizing their importance in the active regulation of cellular function.

Energy Coupling

Coupling to ATP-driven pumps or other high-energy reactions ensures the gradients are stable over time. This maintenance is necessary as, without constant energy input, ions would reach equilibrium distribution, and vital biological processes relying on these gradients would halt. The Gibbs free energy difference across a membrane is a measure of the energy required to maintain these ion gradients, illustrating how biochemical and electrical elements interplay in living systems.