Calculate the free energy of hydrolysis of ATP in a rat liver cell in which the ATP, ADP, and \(P\), concentrations are \(3.4,1.3,\) and \(4.8 \mathrm{m} M\) respectively.

The Gibbs free energy equation is a fundamental tool for predicting the feasibility of reactions in chemistry and biology. It's denoted as \( \Delta G = \Delta G^\circ + RT \ln \frac{{[Products]}}{{[Reactants]}} \), with \( \Delta G^\circ \) being the standard free energy change, R as the universal gas constant, and T as the temperature in Kelvin. In the context of biological reactions such as ATP hydrolysis, the equation takes into account the concentration of reactants and products to determine the actual free energy change (\( \Delta G \)) under non-standard conditions.

Understanding this equation is crucial for assessing whether a reaction will occur spontaneously. A negative \( \Delta G \) means the reaction is exergonic and can proceed without the input of external energy. Conversely, a positive \( \Delta G \) indicates an endergonic reaction requiring energy input to occur. ATP hydrolysis in cellular environments often has a \( \Delta G \) different from the standard value (\( \Delta G^\circ \)), as the actual concentrations of ATP, ADP, and inorganic phosphate (Pi) affect the final free energy of the reaction.

Bioenergetics is the branch of biochemistry that focuses on the energy transformations and transactions in living organisms. It is crucial for understanding how cells harness and use energy to perform various functions such as muscle contraction, nerve conduction, and biosynthesis.

ATP, or adenosine triphosphate, is often referred to as the 'energy currency' of the cell. Its hydrolysis to ADP (adenosine diphosphate) and inorganic phosphate releases energy that is then utilized by the cell. The free energy of ATP hydrolysis is affected by the cell's conditions, including the concentrations of ATP, ADP, and phosphate. Bioenergetics encompasses not only the study of ATP and its role in energy transfer but also the mechanisms by which cells regulate and maintain energy production, storage, and consumption. Understanding these processes is key to grasping how organisms stay alive and function.

Cellular metabolism encompasses the chemical reactions that occur within cells, supporting life by converting nutrients into energy, waste products, and new cellular material. It includes both catabolic pathways, which breakdown molecules to produce energy, and anabolic pathways, which build complex molecules from simpler ones using the energy harnessed from catabolic reactions.

ATP hydrolysis is integrated into numerous metabolic reactions as it provides the necessary energy to drive them. Its role in metabolism is evident in processes such as active transport, where ATP is required to move substances across cell membranes against a concentration gradient. Moreover, the ATP generated during cellular respiration is a critical output of a cell's energy metabolism and is directly correlated with its ability to perform work, grow, and repair itself. Hence, the free energy of ATP hydrolysis is a pivotal factor in the overall functioning of cellular metabolism.