Question

The \(\mathrm{pH}\) of gastric juice is about 1.00 and that of blood plasma is \(7.40 .\) Calculate the Gibbs free energy required to secrete a mole of \(\mathrm{H}^{+}\) ions from blood plasma to the stomach at \(37^{\circ} \mathrm{C}\).

Short answer

The Gibbs free energy required is approximately 37.9 kJ/mol.

Step-by-step solution

Understand the Relationship

To find the Gibbs free energy change (ΔG) for the movement of H⁺ ions from the blood plasma to the stomach, we can use the equation \( \Delta G = -nRT \ln(K) \), where \( n \) is the number of moles of H⁺ ions, \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( K \) is the equilibrium constant of the reaction. For this acid-base reaction, \( K \) can be expressed as \( 10^{\text{pH (stomach)}} / 10^{\text{pH (plasma)}} \).

Convert Temperature to Kelvin

The temperature is given as \(37^{\circ} \mathrm{C}\). Convert this to Kelvin using the formula \( T(\text{K}) = T(^{\circ} \mathrm{C}) + 273.15 \). Thus, \( T = 37 + 273.15 = 310.15 \text{ } \mathrm{K} \).

Calculate the Equilibrium Constant

Using the pH values given, calculate the equilibrium constant \( K \) as follows:\[ K = \frac{10^{1.00}}{10^{7.40}} = 10^{-6.40} \].

Plug in Values into Gibbs Free Energy Equation

Use the equation for Gibbs free energy:\[ \Delta G = -nRT \ln(K) \]For 1 mole of H⁺ ions, \( n = 1 \). Use \( R = 8.314 \text{ } \mathrm{J/mol \cdot K} \), \( T = 310.15 \text{ } \mathrm{K} \), and \( \ln(K) = \ln(10^{-6.40}) = -6.40 \ln(10) \approx -6.40 \times 2.303 \).

Calculate the Gibbs Free Energy

Calculate \( \Delta G \) using the plug-in values:\[ \begin{align*}\Delta G &= -1 \times 8.314 \times 310.15 \times (-6.40 \times 2.303)\&\approx 8.314 \times 310.15 \times 14.7392 \&\approx 37937 \, \text{J/mol} \approx 37.9 \, \text{kJ/mol}.\end{align*} \]

Key concepts

pH

The concept of pH is essential in understanding the acidity or alkalinity of a solution. It is a scale ranging from 0 to 14, with 7 being neutral.
Acids have a pH less than 7, while bases have a pH greater than 7.pH is calculated using the formula:

• \( \text{pH} = -\log[\text{H}^+] \)

In our exercise, the pH of gastric juice is 1.00, indicating a very acidic environment compared to the blood plasma with a pH of 7.40. This difference plays a crucial role in moving H\(^+\) ions between the stomach and blood plasma.Understanding pH helps explain how these ions can affect reactions. Small changes in pH can significantly impact biological processes, especially in acid-base reactions.

Equilibrium Constant

The equilibrium constant, denoted as \( K \), is a critical element in predicting the direction of an acid-base reaction.For any given reaction, \( K \) indicates the proportion of products to reactants at equilibrium. In acid-base reactions involving pH changes, \( K \) can be calculated using the pH values:

• \( K = \frac{10^{\mathrm{pH \; (stomach)}}}{10^{\mathrm{pH \; (plasma)}}} \)

In our context, this results in \( K = 10^{-6.40} \), portraying the ratio of hydrogen ion concentrations between two environments.The equilibrium constant helps us find the Gibbs free energy, as it takes into account the potential energy change when moving ions between different pH environments.

Universal Gas Constant

The universal gas constant, \( R \), is a fundamental constant used in various equations, including the ideal gas law and equations related to chemical thermodynamics. It has a consistent value of 8.314 J/mol·K.In our exercise, \( R \) is used to calculate the Gibbs free energy for moving H\(^+\) ions. It bridges the relationship between energy, moles of gas, and temperature in Kelvin.Using \( R \), we can express energy changes in processes such as ion transport and chemical reactions, making it a vital constant for understanding reaction dynamics.

Temperature Conversion

Temperature plays a significant role in chemical reactions, influencing reaction rates and equilibrium.Converting temperature from Celsius to Kelvin is essential in thermodynamics because most equations are based on the Kelvin scale. The formula is:

• \( T(K) = T(^{\circ}C) + 273.15 \)

In our example, converting 37°C results in 310.15 K.
This conversion is necessary for accurate calculations of Gibbs free energy, as temperature must be in Kelvin units to conform to the SI unit system used in physical chemistry.

Acid-Base Reaction

Acid-base reactions involve the transfer of hydrogen ions (H\(^+\)) between molecules. In this exercise, we deal with the movement of H\(^+\) ions from blood plasma to the stomach, which is a typical acid-base reaction.The main driving force in this process is the difference in pH between the two environments.These reactions are vital in biological systems, influencing functions such as digestion and maintaining pH homeostasis. Understanding these reactions helps unpack how small shifts in hydrogen ion concentration can lead to significant energy changes, as seen in Gibbs free energy calculations.