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Magazine Chapter 15: Problem 50
What size \(\mathrm{pH}\) gradient (the difference between \(\mathrm{pH}_{\text {matrix }}\) and \(\mathrm{pH}_{\text {cytasol }}\) ) would correspond to a free energy change of \(19.2 \mathrm{~kJ} \cdot \mathrm{mol}^{-1}\) ? Assume that \(\Delta \Psi=170 \mathrm{mV}\) and \(T=37^{\circ} \mathrm{C}\).
Short Answer
Expert verified
The pH gradient is approximately 0.81.
Step by step solution
01
Understand the Relationship
The free energy change for moving protons across a membrane can be calculated using the potential difference and pH gradient. The Nernst equation for free energy change due to proton gradient is given by:\[\Delta G = (RT \ln 10) (\Delta pH) + nF\Delta \Psi\]Where \( R \) is the gas constant \( 8.314 \ \text{J} \cdot \text{mol}^{-1} \cdot \text{K}^{-1} \), \( T \) is the temperature in Kelvin, \( \Delta pH \) is the pH gradient, \( n \) is the number of moles of electrons (which is 1 for protons), \( F \) is the Faraday's constant \( 96485 \ \text{C} \cdot \text{mol}^{-1} \), and \( \Delta \Psi \) is the membrane potential in volts.
02
Convert Temperature to Kelvin
Convert the given temperature from Celsius to Kelvin using the formula:\[T(K) = T(\degree C) + 273.15\]Plug in the values:\[T(K) = 37 + 273.15 = 310.15 \text{ K}\]
03
Rearrange for pH Gradient
The main question asks for the pH gradient (\( \Delta pH \)), so rearrange the free energy equation:\[\Delta pH = \frac{\Delta G - nF\Delta \Psi}{RT \ln 10}\]
04
Plug in Known Values
Substitute the given and calculated values into the rearranged equation:\[\Delta pH = \frac{19,200 \ \text{J} \cdot \text{mol}^{-1} - 1 \times 96485 \ \text{C} \cdot \text{mol}^{-1} \times 0.170 \ \text{V}}{8.314 \ \text{J} \cdot \text{mol}^{-1} \cdot \text{K}^{-1} \times 310.15 \ \text{K} \times \ln 10}\]First, calculate the energy due to membrane potential:\[1 \times 96485 \times 0.170 = 16402.45 \ \text{J} \cdot \text{mol}^{-1}\]Now plug this into the full equation:\[\Delta pH = \frac{19,200 - 16402.45}{8.314 \times 310.15 \times \ln 10}\]
05
Calculate Final pH Gradient
Calculate the numerical value:\[\Delta pH = \frac{19,200 - 16402.45}{8.314 \times 310.15 \times 2.302}\]Calculate the denominator:\[8.314 \times 310.15 \times 2.302 \approx 5933.57\]Finally, find \( \Delta pH \):\[\Delta pH = \frac{4797.55}{5933.57} \approx 0.81\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Nernst Equation
The Nernst equation is essential for understanding electrochemistry, as it relates the potential difference across a membrane to the concentration difference of ions on either side. This equation is particularly important when calculating the free energy change due to a pH gradient.
In this context, the Nernst equation is expressed as:
In this context, the Nernst equation is expressed as:
- \( \Delta G = (RT \ln 10) (\Delta pH) + nF\Delta \Psi \)
- \( R \) is the gas constant \( 8.314 \ \text{J} \cdot \text{mol}^{-1} \cdot \text{K}^{-1} \)
- \( T \) is the temperature in Kelvin
- \( n \) is the number of moles of electrons
- \( F \) is Faraday鈥檚 constant \( 96485\ \text{C} \cdot \text{mol}^{-1} \)
- \( \Delta \Psi \) is the membrane potential in volts
Free Energy Change
Free energy change \( (\Delta G) \) tells us how much energy is available to do work during a chemical reaction or a transport process. For biological membranes, it is important to know how much energy is involved in moving protons across membranes.
The formula used in the Nernst equation helps us understand this relation by considering both the pH gradient and the membrane potential. To compute \( \Delta G \), you need both of these factors:
The formula used in the Nernst equation helps us understand this relation by considering both the pH gradient and the membrane potential. To compute \( \Delta G \), you need both of these factors:
- The potential difference, which is affected by \( \Delta \Psi \)
- The difference in proton concentration, represented by \( \Delta pH \)
Membrane Potential
Membrane potential \( (\Delta \Psi) \) is a key concept in cell physiology, referring to the voltage difference across a cell membrane. This potential difference arises due to the distribution of ions on either side of the membrane, often involving protons (\( H^+ \)) in pH-related processes.
In our calculations, the potential is given as \( 170 \ \text{mV} \), or \( 0.170 \ \text{V} \), and plays a significant role in determining the free energy change \( (\Delta G) \) for proton transport.
The process involves:
In our calculations, the potential is given as \( 170 \ \text{mV} \), or \( 0.170 \ \text{V} \), and plays a significant role in determining the free energy change \( (\Delta G) \) for proton transport.
The process involves:
- Movement of charges across membranes
- Contributing to various cellular processes
Temperature Conversion
Temperature conversion is a simple but crucial step in calculations involving thermodynamics, as physical constants are temperature-dependent.
In this exercise, the temperature was provided in Celsius and needed to be converted to Kelvin for accurate calculations. The conversion is straightforward:
In this exercise, the temperature was provided in Celsius and needed to be converted to Kelvin for accurate calculations. The conversion is straightforward:
- Use the formula: \( T(\text{K}) = T(\degree C) + 273.15 \)
- Converting \( 37^{\circ} \text{C} \) gives \( 310.15 \text{ K} \)
