Question

Cells use the hydrolysis of adenosine triphosphate, abbreviated as ATP, as a source of energy. Symbolically, this reaction can be written as $$ \mathrm{ATP}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{ADP}(a q)+\mathrm{H}_{2} \mathrm{PO}_{4}^{-}(a q) $$ where ADP represents adenosine diphosphate. For this reaction, \(\Delta G^{\circ}=-30.5 \mathrm{~kJ} / \mathrm{mol}\). a. Calculate \(K\) at \(25^{\circ} \mathrm{C}\). b. If all the free energy from the metabolism of glucose $$ \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(s)+6 \mathrm{O}_{2}(g) \longrightarrow 6 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(l) $$ goes into forming ATP from ADP, how many ATP molecules can be produced for every molecule of glucose?

Short answer

The equilibrium constant (K) for the ATP hydrolysis reaction at 25°C is approximately \(1.66 \times 10^{11}\). Approximately 92 ATP molecules can be produced from the metabolism of one glucose molecule, assuming all the free energy obtained is used for ATP synthesis.

Step-by-step solution

Write down the formula for calculating the equilibrium constant

We will use the relation between the standard free energy change (∆G°) and the equilibrium constant (K) for a given reaction: \[ \Delta G^\circ = -RT \ln K, \] where R is the ideal gas constant (8.314 J/mol·K), T is the temperature in Kelvin, and ln is the natural logarithm.

Convert the temperature to Kelvin and substitute the given values

First, convert the temperature from Celsius to Kelvin: T = 25°C + 273.15 = 298.15 K. Now, substitute the given values of ∆G° (-30.5 kJ/mol) and R (8.314 J/mol·K) into the formula: \[ -30500 \text{ J/mol} = -(8.314 \text{ J/mol·K})(298.15 \text{ K}) \ln K \]

Solve for K

Divide both sides by (-8.314 J/mol·K)(298.15 K), and then take the exponent of the result to calculate K: \[ K = e^\frac{30500 \text{ J/mol}}{(8.314 \text{ J/mol·K})(298.15 \text{ K})} \approx 1.66 \times 10^{11} \] The equilibrium constant (K) for the ATP hydrolysis reaction at 25°C is approximately \(1.66 \times 10^{11}\). #b. Calculate the number of ATP molecules produced from one glucose molecule#

Calculate the free energy change for the glucose metabolism reaction

The free energy change for the glucose metabolism reaction (∆G_g) can be determined using the standard free energy of formation (∆G_f°) values for the reactants and products in the reaction: \[ \Delta G_g = \sum [\Delta G_f°(\text{products})] - \sum [\Delta G_f°(\text{reactants})] \] ∆G_f° values for the reactants and products are as follows: - Glucose (∆G_f°) = -910 kJ/mol - O2 (∆G_f°) = 0 kJ/mol - CO2 (∆G_f°) = -394 kJ/mol - H2O (∆G_f°) = -228 kJ/mol Using these values, calculate ∆G_g: \[ \Delta G_g = [6(-394 \text{ kJ/mol}) + 6(-228 \text{ kJ/mol})] - [-910 \text{ kJ/mol}] = -2808 \text{ kJ/mol} \] The free energy change for the glucose metabolism reaction is -2808 kJ/mol.

Calculate the number of ATP molecules formed from one glucose molecule

Assuming all the free energy obtained from glucose metabolism is used for the synthesis of ATP from ADP, we can calculate the number of ATP molecules formed by dividing the total free energy change of glucose metabolism by the free energy change of the ATP hydrolysis reaction: \[ \text{Number of ATP molecules} = \frac{\Delta G_g}{\Delta G_{ \text{ATP hydrolysis}}} = \frac{-2808 \text{ kJ/mol}}{-30.5 \text{ kJ/mol}} = 92 \text{ ATP molecules} \] Approximately 92 ATP molecules can be produced from the metabolism of one glucose molecule, assuming all the free energy obtained is used for ATP synthesis.

Key concepts

Free Energy Change

The concept of free energy change, denoted as \(\Delta G^{\circ}\), is crucial for understanding how reactions occur and predict their spontaneity. Free energy change measures the amount of energy available to do work during a chemical reaction.
If \(\Delta G^{\circ} < 0\), the reaction is spontaneous and will proceed without any additional energy input. If \(\Delta G^{\circ} > 0\), the reaction is non-spontaneous and requires energy to occur. For ATP hydrolysis, the \(\Delta G^{\circ}\) is -30.5 kJ/mol, indicating that the reaction releases energy, making it a spontaneous process.
Understanding free energy change is essential in cellular processes as it dictates whether biochemical reactions can occur and how they provide energy for cellular activities.

Equilibrium Constant

The equilibrium constant, symbolized by \(K\), is a vital parameter that describes the ratio of product concentrations to reactant concentrations at equilibrium. This constant helps us understand the extent to which a reaction proceeds.
For ATP hydrolysis, we calculate \(K\) using the equation: \(\Delta G^{\circ} = -RT \ln K\), where \(R\) is the gas constant (8.314 J/mol·K) and \(T\) is the temperature in Kelvin. Through calculations, we find that \(K\) is approximately \(1.66 \times 10^{11}\) at 25°C, indicating that the reaction heavily favors the formation of products at equilibrium.
High values of \(K\) suggest that at equilibrium, the concentration of products will be much greater relative to reactants, highlighting the efficiency of ATP hydrolysis as an energy-transferring mechanism in cells.

Glucose Metabolism

Glucose metabolism is a fundamental biochemical process that provides energy to living organisms, mainly through cellular respiration. This process converts glucose and oxygen into carbon dioxide and water, releasing energy stored in glucose molecules.
In biochemical terms, glucose metabolism is characterized by a considerable free energy change of -2808 kJ/mol, which indicates a substantial release of energy. This energy is then harnessed to produce ATP, which cells use as an energy currency.
During this metabolism, the efficiency of converting energy can be quantified by how many ATP molecules are formed. It's found that one glucose molecule can theoretically produce about 92 ATP molecules, making glucose metabolism highly effective in driving cellular functions.

Adenosine Diphosphate (ADP)

Adenosine Diphosphate, or ADP, plays a crucial role in the energy transfer processes in cells and is a key player in the biochemical cycle of ATP. ADP consists of two phosphate groups and serves as a precursor to ATP.
When the third phosphate group is added to ADP, it forms ATP in a process called phosphorylation.

• This reaction can store energy and is endothermic, meaning it requires an input of energy.
• In contrast, the conversion of ATP back to ADP releases energy, which cells use for various activities.

This cycle of ATP formation from ADP and its subsequent breakdown back to ADP is essential for maintaining the energy balance in living organisms, driving many biological processes, such as muscle contraction, active transport, and biosynthesis.