Given these parameters, what is the ΔG for K+ ion transport into the frog muscle cell in the presence of Ach? ΔG = _ kJ/mol
Dr. Knowsalot smiles and nods. “Good,” she says. “Now let’s see whether K+ ions pass through the open Ach receptors into the cell. As you might remember, these channels are permeable to Na, K+, and Ca²+. For a typical frog muscle cell, the concentration of K+ is roughly 124 mM inside the cell and 2.30 mM outside the cell.” Remember, to calculate ΔG for an ion, you simply add the equations for the chemical and electrostatic components together. ΔG = RT in (C2/C1) + ZFVm. As mentioned earlier, a typical animal cell resting membrane potential is –60 mV, and this process occurs at body temperature, which is 310 K.
The Correct Answer and Explanation is:
To calculate the change in Gibbs free energy (ΔG) for the transport of K⁺ ions into the frog muscle cell in the presence of acetylcholine (Ach), we will use the formula:
[
\Delta G = RT \ln\left(\frac{C_2}{C_1}\right) + ZFV_m
]
Where:
- ( R ) is the gas constant, which is 8.314 J/mol·K.
- ( T ) is the temperature in Kelvin (310 K, as given).
- ( C_1 ) is the concentration of K⁺ outside the cell (2.30 mM).
- ( C_2 ) is the concentration of K⁺ inside the cell (124 mM).
- ( Z ) is the charge of the ion, which for K⁺ is +1.
- ( F ) is the Faraday constant, which is 96,485 C/mol.
- ( V_m ) is the membrane potential, which is given as -60 mV (-0.060 V in volts).
Step 1: Calculate the chemical component (( \Delta G_{\text{chem}} ))
This part of the equation involves the concentration gradient and is calculated using the formula:
[
\Delta G_{\text{chem}} = RT \ln\left(\frac{C_2}{C_1}\right)
]
Substitute the known values:
[
\Delta G_{\text{chem}} = (8.314 \, \text{J/mol·K}) \times (310 \, \text{K}) \times \ln\left(\frac{124 \, \text{mM}}{2.30 \, \text{mM}}\right)
]
[
\Delta G_{\text{chem}} = (8.314 \times 310) \times \ln(53.91)
]
[
\Delta G_{\text{chem}} = 2577.34 \times 3.986
]
[
\Delta G_{\text{chem}} = 10244.96 \, \text{J/mol} = 10.245 \, \text{kJ/mol}
]
Step 2: Calculate the electrostatic component (( \Delta G_{\text{electro}} ))
This part involves the membrane potential and is calculated using:
[
\Delta G_{\text{electro}} = ZFV_m
]
Substitute the known values:
[
\Delta G_{\text{electro}} = (1) \times (96,485 \, \text{C/mol}) \times (-0.060 \, \text{V})
]
[
\Delta G_{\text{electro}} = -5,789.1 \, \text{J/mol} = -5.789 \, \text{kJ/mol}
]
Step 3: Calculate the total ( \Delta G )
Now, we add the two components:
[
\Delta G = \Delta G_{\text{chem}} + \Delta G_{\text{electro}} = 10.245 \, \text{kJ/mol} + (-5.789 \, \text{kJ/mol})
]
[
\Delta G = 4.456 \, \text{kJ/mol}
]
Conclusion:
The ΔG for K⁺ ion transport into the frog muscle cell in the presence of Ach is 4.46 kJ/mol.
Explanation:
The ΔG represents the free energy change for K⁺ transport, accounting for both the concentration gradient (chemical component) and the electrostatic influence (electrostatic component). The concentration gradient favors the movement of K⁺ into the cell (positive ΔG from chemical gradient), but the membrane potential works against this movement (negative ΔG from electrostatic gradient). Since the total ΔG is positive, the transport of K⁺ into the cell is not spontaneous under the given conditions.