Question

What is the maximum amount of work that can be obtained from hydrolyzing 1 mole of ATP to ADP and phosphate at \(298 \mathrm{~K}\) ? Assume that the concentrations of all reactants are \(0.01 \mathrm{M}\) and \(\Delta G^{\circ}\) is \(-32.5 \mathrm{~kJ} / \mathrm{mol}\). If the conversion of free energy to mechanical work is \(100 \%\) efficient, how many moles of ATP would have to be hydrolyzed to lift a \(100 \mathrm{~kg}\) weight 1 meter high?

Short answer

Maximum work from 1 mole ATP is 44.4 kJ; 0.022 moles ATP lifts the weight.

Step-by-step solution

Determine the reaction quotient (Q)

Using the given concentrations of reactants, calculate the reaction quotient (Q) of the hydrolysis reaction: \[ Q = \frac{{\text{{[ADP]}} \times \text{{[Pi]}}}}{{\text{{[ATP]}}}} = \frac{{0.01 \times 0.01}}{{0.01}} = 0.01 \] where [ADP], [Pi], and [ATP] are the concentrations of ADP, phosphate (Pi), and ATP respectively.

Calculate the actual Gibbs free energy (ΔG)

Use the equation that relates standard Gibbs energy change and the reaction quotient: \[ \Delta G = \Delta G^{\circ} + RT \ln(Q) \] Here, \( \Delta G^{\circ} = -32.5 \mathrm{~kJ/mol} \), \( R = 8.314 \times 10^{-3} \mathrm{~kJ/(mol \cdot K)} \), and \( T = 298 \mathrm{~K} \). Thus, \[ \Delta G = -32.5 + (8.314 \times 10^{-3} \times 298 \times \ln(0.01)) \approx -32.5 - 11.9 \approx -44.4 \mathrm{~kJ/mol} \]

Calculate the work for lifting the weight

The work done to lift a weight is given by \( W = mgh \). Given that \( m = 100 \mathrm{~kg} \), \( g = 9.8 \mathrm{~m/s^2} \), and \( h = 1 \mathrm{~m} \), the work required is: \[ W = 100 \times 9.8 \times 1 = 980 \mathrm{~J} = 0.98 \mathrm{~kJ} \]

Calculate number of moles of ATP required

Since the conversion of free energy to work is 100% efficient, the maximum work obtainable per mole of ATP is \(-\Delta G = 44.4 \mathrm{~kJ/mol}\). Divide the work required by the work obtainable per mole of ATP: \[ \text{{Moles of ATP}} = \frac{{0.98 \mathrm{~kJ}}}{{44.4 \mathrm{~kJ/mol}}} \approx 0.022 \text{{ moles}} \]

Conclusion: Calculate the maximum work and required ATP

The maximum work obtainable from hydrolyzing 1 mole of ATP is \( 44.4 \mathrm{~kJ} \). To lift a 100 kg weight 1 meter requires roughly \( 0.022 \) moles of ATP.

Key concepts

ATP Hydrolysis

ATP, which stands for adenosine triphosphate, is often referred to as the energy currency of the cell. ATP hydrolysis is the process of breaking down ATP into adenosine diphosphate (ADP) and inorganic phosphate (Pi). This reaction releases energy that can be used by cells to perform various functions. At physiological conditions, ATP undergoes hydrolysis in the presence of water, resulting in the release of a significant amount of energy.

• The reaction is typically represented as: ATP + H₂O → ADP + Pi + Energy.
• The standard change in free energy (ΔG°) for this reaction is often negative, showing that it is exergonic and releases energy.

The reaction is spontaneous under standard conditions due to the high energy stored in the phosphate bonds of ATP. By understanding ATP hydrolysis, you can see how cells power various processes, like muscle contraction and synthesis of biomolecules.
In practical settings, medical and biological research utilizes ATP hydrolysis to explore enzyme functions and cellular energy management.

Gibbs Free Energy

Gibbs free energy is a concept in thermodynamics that helps us understand and predict the feasibility of a chemical reaction. Gibbs energy combines two important thermodynamic properties: enthalpy (heat content) and entropy (degree of disorder). The equation is given by:\[ G = H - TS \]where:

• \( G \) is the Gibbs free energy,
• \( H \) is the enthalpy,
• \( T \) is the temperature in Kelvin, and
• \( S \) is the entropy.

If ΔG (change in Gibbs free energy) is negative, the reaction is spontaneous, meaning it can occur without needing external energy input. In scenarios like ATP hydrolysis, a negative ΔG indicates that the reaction will release energy that can be harnessed efficiently.
By calculating ΔG under specific conditions, you determine the actual energy available in a system, like in the given problem where hydrolyzing ATP resulted in a ΔG of -44.4 kJ/mol, highlighting available energy for work conversion.

Mechanical Work Conversion

In the context of thermodynamics and ATP hydrolysis, mechanical work conversion involves taking the energy released from chemical reactions and converting it into physical movement. In the original exercise about lifting a weight with energy from ATP, this concept is a practical application.When the energy from ATP hydrolysis is used to do mechanical work, like lifting an object against Earth's gravity, the efficiency of this conversion is crucial. In ideal conditions (100% efficient), all the free energy from ATP is converted into mechanical work. For instance, the energy to lift a 100 kg weight 1 meter high is calculated as follows:\[ W = mgh \]where:

• \( W \) is the work done,
• \( m \) is the mass of the object,
• \( g \) is the acceleration due to gravity, and
• \( h \) is the height.

By converting kJ to Joules and estimating the ATP moles needed, we can see the practical energy conversion. This showcases how ATP fuels muscular activity, providing vital insights for fields such as sports science and metabolic research.
Understanding mechanical work conversion emphasizes the importance of energy balance and efficiency in biological systems.