Question

Energy for H+Pumping The parietal cells of the stomach lining contain membrane "pumps" that transport hydrogen ions from the cytosol (pH7.0) into the stomach, contributing to the acidity of gastric juice (pH 1.0). Calculate the free energy required to transport 1 mol of hydrogen ions through these pumps. (Hint: See Chapter 11.) Assume a temperature of 37∘C.

Answer 1

The free energy change required to transport 1 mol of hydrogen ions through these pumps is approximately -35,600 J/mol. The negative sign indicates that the process is exergonic, meaning it releases energy.

To calculate the free energy change
`(\(\Delta G\))` required to transport 1 mol of hydrogen ions through these pumps, you can use the Nernst equation. The Nernst equation relates the free energy change to the concentration difference of the ions across a membrane. The equation is given by:

`\[ \Delta G = -RT \ln\left(\frac{[H^+]_{\text{final}}}{[H^+]_{\text{initial}}}\right) \]`

Where:

• 
  `\(\Delta G\)` is the free energy change,
• 
  `\(R\)` is the gas constant
  `(\(8.314 \, \text{J} \cdot \text{mol}^(-1) \cdot \text{K}^(-1)\)),`
• 
  `\(T\)` is the temperature in Kelvin (37 °C = 310 K),
• 
  `\([H^+]_{\text{final}}` is the final concentration of hydrogen ions (pH 1.0), and
• 
  `\([H^+]_{\text{initial}}\)` is the initial concentration of hydrogen ions (pH 7.0).

First, convert the pH values to hydrogen ion concentrations using the formula:
`\([H^+] = 10^{-\text{pH}}`.

`\[ [H^+]_{\text{final}} = 10^{-\text{pH}_{\text{final}}}`

`\ [H^+]_{\text{initial}} = 10^{-\text{pH}_{\text{initial}}}`

Substitute these values into the Nernst equation and solve for
`\(\Delta G\)`:

`\[ \Delta G = -8.314 \, \text{J} \cdot \text{mol}^(-1) \cdot \text{K}^(-1) \cdot 310 \, \text{K} \cdot \ln\left((10^(-1.0))/(10^(-7.0))\right) \]`

Calculate the numerical value to get the final result for
`\(\Delta G\)`.

Let's calculate it step by step:

Convert pH values to hydrogen ion concentrations:

`\[ [H^+]_{\text{final}} = 10^{-\text{pH}_{\text{final}}} = 10^(-1.0) \]`

`\[ [H^+]_{\text{initial}} = 10^{-\text{pH}_{\text{initial}}} = 10^(-7.0) \]`

Substitute these values into the Nernst equation:

`\[ \Delta G = -8.314 \, \text{J} \cdot \text{mol}^(-1) \cdot \text{K}^(-1) \cdot 310 \, \text{K} \cdot \ln\left((10^(-1.0))/(10^(-7.0))\right) \]`

Calculate the argument of the natural logarithm:

`\[ \ln\left((10^(-1.0))/(10^(-7.0))\right) = \ln(10^6) = 6 \ln(10) \]`

Use the fact that
`\(\ln(10) \approx 2.3026\)`:

`\[ \ln(10) \approx 2.3026 \]`

`\[ 6 \ln(10) \approx 6 * 2.3026 \]`

Multiply the result by the remaining constants:

`\[ \Delta G = -8.314 \, \text{J} \cdot \text{mol}^(-1) \cdot \text{K}^(-1) \cdot 310 \, \text{K} \cdot (6 * 2.3026) \]`

`\[ \Delta G \approx -8.314 \, \text{J} \cdot \text{mol}^(-1) \cdot \text{K}^(-1) \cdot 310 \, \text{K} \cdot 13.8156 \]`

Calculate the numerical value:

`\[ \Delta G \approx - 8.314 \, \text{J} \cdot \text{mol}^(-1) \cdot \text{K}^(-1) \cdot 4281.756 \]`

`\[ \Delta G \approx -35,600 \, \text{J/mol} \]`

Answer 2

The free energy required to transport 1 mol of hydrogen ions through the pumps is approximately -179,549,000 J/mol.

The negative value shows that the process is energetically favorable and requires energy input from the cell, likely from ATP hydrolysis.

ΔpH = pH_in - pH_out = 7.0 - 1.0 = 6.0

Δpconc = 1 μM / 1 M = 1e-6

ΔG = F * T * ln(Δpconc)

ΔG = 96,485 C/mol * 310.15 K * ln(1e-6)

ΔG = -179,549 kJ/mol

ΔG= -179,549,000 J/mol

ΔG_total = ΔG * 1 mol

= -179,549,000 J/mol