Question

Energetics of Phototransduction During photosynthesis, pigment molecules in chloroplasts must absorb eight photons (four by each photosystem) for every \(\mathrm{O}_{2}\) molecule they produce, according to the equation $$ 2 \mathrm{H}_{2} \mathrm{O}+2 \mathrm{NADP}^{+}+8 \text { photons } \rightarrow 2 \mathrm{NADPH}+2 \mathrm{H}^{+}+\mathrm{O}_{2} $$ The \(\Delta G^{\prime \circ}\) for the light-independent production of \(\mathrm{O}_{2}\) is 400 \(\mathrm{kJ} / \mathrm{mol}\). Assuming that these photons have a wavelength of \(700 \mathrm{~nm}\) (red) and that the light absorption and use of light energy are \(100 \%\) efficient, calculate the free- energy change for the process.

Short answer

The net free-energy change for the process is 969.6 kJ/mol.

Step-by-step solution

Find the Energy of One Photon

To find the energy of one photon, we use the formula:\[ E = \frac{hc}{\lambda} \]where:- \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ Js}) \)- \( c \) is the speed of light \( (3 \times 10^8 \text{ m/s}) \)- \( \lambda \) is the wavelength of the photon (700 nm or \( 700 \times 10^{-9} \) m)Plug in these values:\[ E = \frac{(6.626 \times 10^{-34} \text{ Js})(3 \times 10^8 \text{ m/s})}{700 \times 10^{-9} \text{ m}} \]\[ E = 2.84 \times 10^{-19} \text{ J/photon} \]This is the energy of one photon with a wavelength of 700 nm.

Convert Photon Energy to kJ/mol

We need the energy in kJ/mol to compare with \( \Delta G^{\prime \circ} \). First, convert the energy per photon from Joules to kilojoules. There are \( 6.022 \times 10^{23} \) photons in a mole (Avogadro's number).Convert Joules to kilojoules:\[ 2.84 \times 10^{-19} \text{ J/photon} \times \frac{1 \text{ kJ}}{1000 \text{ J}} = 2.84 \times 10^{-22} \text{ kJ/photon} \]Then multiply by Avogadro's number to get the energy per mole:\[ 2.84 \times 10^{-22} \text{ kJ/photon} \times 6.022 \times 10^{23} \text{ photons/mol} = 171.2 \text{ kJ/mol} \]This is the energy of one mole of photons with a wavelength of 700 nm.

Calculate Total Energy from Eight Photons

Since the process requires 8 photons per molecule of \( \text{O}_2 \), multiply the energy per mole of photons by 8:\[ 171.2 \text{ kJ/mol} \times 8 = 1369.6 \text{ kJ/mol} \]Thus, the total free energy obtained from absorbing 8 photons is 1369.6 kJ/mol.

Determine the Net Free-Energy Change

Given that \( \Delta G^{\prime \circ} \) for the production of \( \text{O}_2 \) is 400 kJ/mol, if the photon absorption and use are 100% efficient, the net free-energy change for the process is equal to the total energy obtained from photons minus the energy required:\[ \Delta G = 1369.6 \text{ kJ/mol} - 400 \text{ kJ/mol} = 969.6 \text{ kJ/mol} \]This represents the net free-energy change available for storage or use in other processes.

Key concepts

Photosynthesis

Photosynthesis is a fundamental process by which green plants, algae, and some bacteria convert light energy into chemical energy, stored in glucose, a sugar molecule. This process occurs in two main stages: light-dependent reactions and light-independent reactions (Calvin Cycle). During the light-dependent reactions, chloroplasts within plant cells capture light energy. This initiation phase involves the absorption of photons, facilitating the transfer of energy through various pigments. These pigments include chlorophyll a, chlorophyll b, and carotenoids, each playing a unique role in capturing different wavelengths of light.

The absorbed light energy is then used to split water molecules, releasing oxygen (O₂) as a byproduct. This split also produces protons and electrons that help form energy-rich molecules like ATP (adenosine triphosphate) and NADPH (reduced nicotinamide adenine dinucleotide phosphate). These molecules are vital as they provide the energy and electrons required for the next stage of photosynthesis.

In the Calvin Cycle, carbon dioxide (CO₂) is fixed into organic molecules that eventually form glucose. This cycle does not require light directly but relies on the products of the light reactions (ATP and NADPH) to drive the synthesis of sugar from CO₂.

Chloroplasts

Chloroplasts are specialized organelles within plant cells where photosynthesis takes place. They contain a unique system of membranes called thylakoids, arranged into stacks known as grana. These thylakoid membranes house the pigment molecules that capture light energy, starting the process of photosynthesis.

• Structure: Chloroplasts are surrounded by a double membrane, consisting of an outer membrane and an inner membrane, which encloses the stroma. The stroma is a semi-fluid substance where the Calvin Cycle occurs.
• Functions: Within the thylakoid membranes, chlorophyll and other pigments absorb light, which drives the formation of ATP and NADPH during the light-dependent reactions. The chloroplast's structure is optimized to maximize its exposure to sunlight, enhancing its efficiency in capturing and converting light energy.

Chloroplasts are pivotal because they convert solar energy into a useable form, supporting life on Earth by contributing to the oxygen supply and forming the base of the food chain.

Photon Energy

Photon energy refers to the energy carried by a single photon, such as those part of light. It is a critical concept in understanding photosynthesis, as photons are the fundamental units of light energy absorbed by chloroplasts.

The energy of a photon depends on its wavelength and is calculated using the equation:\[ E = \frac{hc}{\lambda} \]where:

• \( E \) is the energy of the photon,
• \( h \) is Planck's constant \((6.626 \times 10^{-34} \text{ J \cdot s})\),
• \( c \) is the speed of light \((3 \times 10^8 \text{ m/s})\),
• \( \lambda \) is the wavelength of the photon.

In photosynthesis, the energy captured from photons through chlorophyll and other pigments is crucial for driving the chemical reactions that convert carbon dioxide and water into glucose and oxygen.

Each absorbed photon excites a chlorophyll molecule, which subsequently initiates electron transport through the photosystems, ultimately resulting in the formation of ATP and NADPH.

Free Energy Calculation

Free energy calculation is essential for understanding the energetics involved in photosynthesis. It involves calculating the change in free energy (\( \Delta G \)) during a chemical reaction, which indicates whether a reaction can occur spontaneously.

• Free Energy Change (\( \Delta G \)): In photosynthesis, the net free energy change can be calculated by comparing the energy derived from absorbed photons with the energy required to produce a molecule of oxygen. The equation used for calculating \( \Delta G \) is:
  \[ \Delta G = 1369.6 \text{ kJ/mol} - 400 \text{ kJ/mol} \]where 1369.6 kJ/mol is the energy obtained from photons and 400 kJ/mol is the energy needed for oxygen production.
• Significance: A positive \( \Delta G \) value, as obtained here, implies an availability of free energy for storage or further reactions. This underscores the efficiency of photosynthetic systems in not just supporting plant life but in maintaining terrestrial life by producing oxygen and organic matter used as food.

The calculation not only helps in comprehending photosynthesis efficiency but also highlights the balance of energy transformation crucial for life processes.