Thermodynamics of Membrane Diffusion: Charged Ion: Videos & Practice Problems

In membrane diffusion for charged ions, the transmembrane potential (Δψ) is crucial, representing the difference in electrical charge across the membrane. The Gibbs free energy change (ΔG_transport) for ion transport is given by the equation: ${\mathrm{\Delta G}}^{\mathrm{transport}}=zf\mathrm{\Delta \psi }$, where z is the net charge of the ion and f is Faraday's constant. The electrochemical gradient combines both chemical and electrical gradients, essential for understanding ion diffusion across membranes.

Thermodynamics of Membrane Diffusion: Charged Ion

In the study of thermodynamics related to membrane diffusion, it is essential to differentiate between uncharged molecules and charged ions. When focusing on charged ions, particularly cations, the transmembrane potential, denoted as Δψ, plays a crucial role. This potential represents the difference in electrical charge across the membrane and must be considered when analyzing the diffusion of charged particles.

The transport free energy change for ions, ΔG_transport, can be expressed by the equation:

ΔG_transport = Z × F × Δψ

In this equation, Z represents the net charge of the diffusing ion, while F is Faraday's constant, which quantifies the charge of one mole of electrons. Faraday's constant is approximately 96,485 C/mol (coulombs per mole), and it is always expressed as a positive value, despite the negative charge of electrons. Understanding the units of Faraday's constant is important; it can be represented as J/V·mol (joules per volt per mole) or C/mol.

However, the diffusion of ions is not solely influenced by the electrical gradient. It is essential to consider the electrochemical gradient, which combines both the chemical gradient and the electrical gradient. The chemical gradient portion of the equation remains the same as that for uncharged molecules, while the electrical gradient is added for charged ions. Thus, the complete equation for the thermodynamics of membrane diffusion for charged ions incorporates both gradients:

ΔG_transport = ΔG_chemical + Z × F × Δψ

This comprehensive approach allows for a more accurate understanding of how charged ions diffuse across membranes, taking into account both the concentration differences and the electrical forces at play. In future applications, this equation will be utilized to calculate specific scenarios involving the thermodynamics of membrane diffusion for charged ions.

Thermodynamics of Membrane Diffusion: Charged Ion Example 1

To calculate the energy cost of transporting calcium ions (Ca²⁺) from the cytosol to the extracellular space, we follow a systematic four-step process. This example considers a temperature of 37 degrees Celsius, a transmembrane potential (ΔΨ) of 0.05 volts, a cytosolic calcium concentration of 1.0 × 10^-7 M, and an extracellular calcium concentration of 1.0 × 10^-3 M.

In the first step, we identify the net charge of the diffusing molecule. Calcium ions carry a charge of +2, which indicates that we will use an equation that accounts for both the chemical and electrical gradients, as charged ions diffuse based on the electrochemical gradient.

Step two involves determining the direction of diffusion and the sign of the transmembrane potential. In this case, calcium is pumped from the cytosol (initial side) to the extracellular space (final side). The transmembrane potential is calculated as ΔΨ = Ψ_out - Ψ_in. Since the inside of the cell is more negative than the outside, ΔΨ is positive, confirming that we can use the given positive value of 0.05 volts in our calculations.

In step three, we check the units of all values to ensure compatibility. The temperature must be converted from degrees Celsius to Kelvin. The conversion is done by adding 273.15, resulting in a temperature of 310.15 K. The concentrations of calcium are already in compatible molarity units, so no changes are needed there.

Finally, in step four, we plug the values into the equation for ΔG_transport:

ΔG_transport = RT ln([C_final]/[C_initial]) + zFΔΨ

Where:

• R = 8.315 J/(mol·K)
• T = 310.15 K
• [C_final] = 1.0 × 10^-3 M
• [C_initial] = 1.0 × 10^-7 M
• z = +2 (net charge of calcium)
• F = 96,485 J/(V·mol) (Faraday's constant)
• ΔΨ = 0.05 V

Calculating the chemical gradient:

ΔG_chemical = (8.315 J/(mol·K) × 310.15 K) ln(1.0 × 10^-3 / 1.0 × 10^-7)

Calculating the electrical gradient:

ΔG_electrical = zFΔΨ = 2 × 96,485 J/(V·mol) × 0.05 V

After performing the calculations, we find:

• ΔG_chemical ≈ 2,578.9 J/mol
• ΔG_electrical ≈ 9,648.5 J/mol

Adding these values gives:

ΔG_transport ≈ 33,400.9 J/mol

To convert this to kilojoules per mole, we divide by 1,000, resulting in:

ΔG_transport ≈ 33.4 kJ/mol

This value represents the energy cost of pumping calcium ions under the specified conditions, concluding the example problem.

Calculate the free energy change (ΔG _transport) for the movement of Na ⁺ into a cell when its concentration outside is 150 mM and its cytosolic concentration is 10 mM. Assume that T = 20°C and ΔΨ = –50 mV (inside negative).

• A. –1.7 KJ/mol.
• B. –11.4 KJ/mol.
• C. –11,600 KJ/mol.
• D. 11.4 KJ/mol.
• E. 14.3 KJ/mol.

Answer: (B)

Calculate the ΔG_transport required to move 1 mole of Na ⁺ ions from inside the cell ([Na⁺] inside = 5 mM) to the outside of the cell ([Na ⁺] outside = 150 mM) when ΔΨ = –70 mV (inside negative) & the temperature is 37°C.

• A. –15.5 KJ/mol.
• B. 2.0 KJ/mol
• C. 15.5 KJ/mol.
• D. –2.0 KJ/mol.

Answer: (C)

Calculate the ΔG_transport when Ca²⁺ ions move from the endoplasmic reticulum ([Ca²⁺] = 1 mM) to the cytoplasm ([Ca²⁺] = 0.1 μM). Assume that ΔΨ = 0 and T = 25°C.

• A. 23 KJ/mol.
• B. –23 KJ/mol.
• C. –17 KJ/mol.
• D. 17 KJ/mol.

Answer: (B)

Here’s what students ask on this topic:

What is the role of the transmembrane potential (Δψ) in the diffusion of charged ions across a membrane?

The transmembrane potential (Δψ) represents the difference in electrical charge across a membrane and is crucial for the diffusion of charged ions. Unlike uncharged molecules, charged ions are influenced by both the chemical gradient and the electrical gradient. The transmembrane potential contributes to the electrochemical gradient, which determines the direction and rate of ion diffusion. The Gibbs free energy change (ΔG_transport) for ion transport is given by the equation:

$\mathrm{\Delta G}$_transport=z⋅f⋅Δψ

where z is the net charge of the ion and f is Faraday's constant. This equation shows how the transmembrane potential directly affects the energy required for ion transport.

How does Faraday's constant (f) factor into the thermodynamics of membrane diffusion for charged ions?

Faraday's constant (f) is a key factor in the thermodynamics of membrane diffusion for charged ions. It represents the magnitude of the charge of one mole of electrons and is always a positive value, approximately 96,485 coulombs per mole. In the context of ion transport, Faraday's constant is used in the equation for the Gibbs free energy change (ΔG_transport):

$\mathrm{\Delta G}$_transport=z⋅f⋅Δψ

Here, z is the net charge of the ion and Δψ is the transmembrane potential. Faraday's constant helps quantify the energy change associated with moving charged ions across a membrane, making it essential for understanding the electrochemical gradient.

What is the difference between the chemical gradient and the electrical gradient in membrane diffusion?

The chemical gradient and the electrical gradient are two components of the electrochemical gradient that influence membrane diffusion. The chemical gradient refers to the concentration difference of a substance across a membrane, driving diffusion from high to low concentration. The electrical gradient, on the other hand, is the difference in electrical charge across the membrane, influencing the movement of charged ions. For charged ions, both gradients must be considered together, forming the electrochemical gradient. The Gibbs free energy change (ΔG_transport) for ion transport combines these gradients:

$\mathrm{\Delta G}$_transport=z⋅f⋅Δψ+RTln(C_final/C_initial)

where z is the ion's charge, f is Faraday's constant, Δψ is the transmembrane potential, R is the gas constant, T is the temperature, and C represents concentrations.

How do you calculate the Gibbs free energy change (ΔG_transport) for the diffusion of charged ions across a membrane?

To calculate the Gibbs free energy change (ΔG_transport) for the diffusion of charged ions across a membrane, you need to consider both the chemical and electrical gradients. The equation is:

$\mathrm{\Delta G}$_transport=z⋅f⋅Δψ+RTln(C_final/C_initial)

where z is the net charge of the ion, f is Faraday's constant (96,485 C/mol), Δψ is the transmembrane potential, R is the gas constant (8.314 J/(mol·K)), T is the temperature in Kelvin, and C represents the ion concentrations on either side of the membrane. This equation combines the effects of the electrical gradient (z·f·Δψ) and the chemical gradient (RT ln(C_final/C_initial)) to determine the total energy change for ion transport.

Why is the electrochemical gradient important for the diffusion of charged ions across membranes?

The electrochemical gradient is crucial for the diffusion of charged ions across membranes because it combines both the chemical and electrical gradients. The chemical gradient is the concentration difference of ions across the membrane, while the electrical gradient is the difference in electrical charge. Together, they determine the direction and rate of ion movement. The Gibbs free energy change (ΔG_transport) for ion transport is given by:

$\mathrm{\Delta G}$_transport=z⋅f⋅Δψ+RTln(C_final/C_initial)

where z is the ion's charge, f is Faraday's constant, Δψ is the transmembrane potential, R is the gas constant, T is the temperature, and C represents concentrations. This equation shows how both gradients influence ion transport, making the electrochemical gradient essential for understanding membrane diffusion.