The standard free energy change for the hydrolysis of ATP was given in Problem 91. In a particular cell, the concentrations of ATP, ADP, and Pi are 0.0031 M, 0.0014 M, and 0.0048 M, respectively. Calculate the free energy change for the hydrolysis of ATP under these conditions. (Assume a temperature of 298 K.)

The concept of standard free energy change, denoted as \( \Delta G^\circ \), is a cornerstone in understanding chemical reactions and energy transfers in biological processes. It represents the change in free energy of a reaction under standard conditions — a pressure of 1 bar, a temperature of 25°C (or 298 K), and all reactants and products at 1 M concentration. In the context of ATP hydrolysis — the breakdown of ATP into ADP and inorganic phosphate (Pi) — this value indicates how much energy is released or consumed during the reaction under these standard conditions.

In biochemistry, a negative \( \Delta G^\circ \) value for a reaction such as ATP hydrolysis suggests the process releases energy and can occur spontaneously. Conversely, a positive value would imply that energy must be input for the reaction to proceed. Knowing the standard free energy change allows scientists and students to predict how a reaction will behave in an ideal setting and serves as a baseline for further calculations in non-standard conditions.

The Gibbs free energy equation is fundamental in the field of chemical thermodynamics, providing a quantitative measure of the spontaneity of a reaction and the maximum amount of work obtainable from a chemical process at constant pressure and temperature. The equation is written as \( \Delta G = \Delta G^\circ + RT\ln\frac{[ADP][Pi]}{[ATP]} \), where \( \Delta G \) is the change in free energy, \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and the terms in brackets represent the molar concentrations of the reactants and products.

The beauty of this formula lies in its ability to account for the actual conditions within a cell or reaction environment. As concentrations deviate from the standard 1 M, the Gibbs equation allows for recalculating the free energy change, offering a more accurate description of the reaction’s potential to do work. This is crucial for biological systems, where concentrations often differ significantly from standard conditions.

The universal gas constant, symbolized as \( R \), is an essential constant in the realm of physical sciences, particularly in the equations of chemical thermodynamics. \( R \) bridges the gap between the macroscopic and microscopic worlds, allowing us to use macroscopic measurements like pressure, volume, and temperature to calculate the behavior of gases on a molecular level. It has a value of 8.314 J/mol∙K and is featured in the ideal gas law as well as the Gibbs free energy equation.

Its role in the Gibbs equation is integral, as it enables the conversion of the natural logarithm of a reaction quotient (which includes concentrations of reactants and products) into energy units that quantify the change in free energy of a system under non-standard conditions. Without \( R \), it would be impossible to accurately determine how changes in temperature and reactant concentrations affect the free energy and spontaneity of a reaction.

Chemical thermodynamics is the study of energy transformations in chemical reactions and the relationships between properties such as temperature, pressure, volume, and concentration. This field is pivotal in understanding how reactions occur, predicting the feasibility of reactions, and designing processes that can harness or require energy changes.

In biological systems, thermodynamics is applied to understand the energy flow during metabolic pathways. ATP hydrolysis, for instance, is a thermodynamic process that powers a wide range of cellular activities. The free energy released by breaking down ATP is used to perform cellular work, from muscle contraction to active transport across membranes. The principles of chemical thermodynamics — embodied in the standard free energy change, Gibbs free energy equation, and the incorporation of constants like \( R \) — are the tools that enable us to analyze such intricate biological processes and design efficient biochemical experiments or medical treatments.