Questions: Calculate the free energy per mole to transport calcium ions up a concentration gradient from 0.002M to 0.015M at 37 Celsius

Solution Steps

The free energy change (\(\Delta G\)) for transporting ions across a concentration gradient is given by the Nernst equation: \[ \Delta G = RT \ln \left( \frac{C_2}{C_1} \right) \] where:

• \(R\) is the universal gas constant (\(8.314 \, \text{J/mol·K}\)),
• \(T\) is the temperature in Kelvin,
• \(C_2\) is the final concentration,
• \(C_1\) is the initial concentration.

The temperature in Celsius is given as 37°C. Convert this to Kelvin: \[ T = 37 + 273.15 = 310.15 \, \text{K} \]

Substitute the given concentrations and the temperature into the Nernst equation: \[ \Delta G = 8.314 \times 310.15 \times \ln \left( \frac{0.015}{0.002} \right) \]

Calculate the natural logarithm: \[ \ln \left( \frac{0.015}{0.002} \right) = \ln (7.5) \approx 2.0149 \]

Now, compute \(\Delta G\): \[ \Delta G = 8.314 \times 310.15 \times 2.0149 \approx 5190.1 \, \text{J/mol} \]

Final Answer