Gastric juice (pH 1.5) is produced by pumping HCl from blood...

Gastric juice (pH 1.5) is produced by pumping HCl from blood plasma (pH 7.4 ) into the stomach. Calculate the amount of free energy required to concentrate the $\mathrm{H}^{+}$ in 1 L of gastric juice at $37^{\circ} \mathrm{C}$. Under cellular conditions, how many moles of ATP must be hydrolyzed to provide this amount of free energy? The free-energy change for ATP hydrolysis under cellular conditions is about $-58 \mathrm{kJ} / \mathrm{mol}$ (as explained in Chapter 13 ). Ignore the effects of the transmembrane electrical potential.

Key Concepts

Gibbs Free Energy

Gibbs free energy is a thermodynamic quantity that indicates the amount of work obtainable from a process at constant temperature and pressure. In the context of ion transport, it quantifies the energy required to move ions against a concentration gradient, using the relationship ?G = RT ln(C_final/C_initial), which allows us to calculate the energy cost per mole of ions being transferred.

Concentration Gradient

A concentration gradient refers to the difference in the concentration of a particular substance between two regions. Energy is required to move a substance such as H+ from an area of lower concentration (or higher pH) to an area of higher concentration (or lower pH). This gradient is central to understanding processes like the active transport of ions, where work is done to concentrate ions in a specific region against their natural tendency to diffuse.

pH Scale and Hydrogen Ion Concentration

The pH scale is a logarithmic measure of the hydrogen ion concentration in a solution. A low pH indicates a high concentration of H+ ions, while a high pH indicates a lower concentration. In this problem, the difference in pH between blood plasma and gastric juice defines the magnitude of the concentration gradient the cell must work against to pump H+ into the stomach.

ATP Hydrolysis

ATP hydrolysis is the process by which adenosine triphosphate (ATP) is broken down into ADP and an inorganic phosphate, releasing energy. This reaction is often used by cells to drive various endergonic (energy-requiring) processes, including the active transport of ions against concentration gradients. The free energy released from ATP hydrolysis under cellular conditions sets the efficiency limit for how much work can be done per mole of ATP, and in this context, it is used to determine the number of ATP molecules required to meet the energy demand for pumping H+ ions.

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