Question

How Often Does a Chlorophyll Molecule Absorb a Photon? The amount of chlorophyll \(a\left(M_{\mathrm{r}} 892\right)\) in a spinach leaf is about \(20 \mu \mathrm{g} / \mathrm{cm}^{2}\) of leaf surface. In noonday sunlight (average energy reaching the leaf is \(5.4 \mathrm{~J} / \mathrm{cm}^{2} \cdot \mathrm{min}\) ), the leaf absorbs about \(50 \%\) of the radiation. How often does a single chlorophyll molecule absorb a photon? Given that the average lifetime of an excited chlorophyll molecule in vivo is 1 ns, what fraction of the chlorophyll molecules are excited at any one time?

Short answer

Each chlorophyll molecule absorbs a photon every 88 ms. About 1.14 x 10^-8 of the chlorophyll molecules are excited at any one time.

Step-by-step solution

Calculate Energy Absorbed by Leaf

First, calculate the energy absorbed by the leaf. The energy reaching the leaf is \(5.4 \, \text{J/cm}^2 \cdot \text{min}\). Since the leaf absorbs 50% of this energy, the absorbed energy is \(0.5 \times 5.4 = 2.7 \, \text{J/cm}^2 \cdot \text{min}\).

Convert Absorbed Energy to Photons

Next, convert the absorbed energy to the number of photons. The energy of a single photon of chlorophyll absorption, corresponding to a wavelength of ~680 nm, is given by \(E = \frac{hc}{\lambda}\), where \(h = 6.63 \times 10^{-34} \text{ J}\cdot\text{s}\), \(c = 3.00 \times 10^8 \text{ m/s}\), and \(\lambda = 680 \times 10^{-9} \text{ m}\). Calculating this gives us \(E \approx 2.93 \times 10^{-19} \text{ J/photon}\). Thus, the number of photons absorbed per \(\text{cm}^2\) per minute is \( \frac{2.7}{2.93 \times 10^{-19}} \approx 9.22 \times 10^{18} \) photons.

Determine Number of Chlorophyll Molecules

Calculate the number of chlorophyll molecules per \(\text{cm}^2\). The amount of chlorophyll is given as \(20 \, \mu\text{g/cm}^2\), which equals \(20 \times 10^{-6} \, \text{g/cm}^2\). The molar mass \(M_r\) of chlorophyll is 892 g/mol. Thus, the number of moles of chlorophyll per \(\text{cm}^2\) is \( \frac{20 \times 10^{-6}}{892} \text{ mol} \). Using Avogadro's number \(6.022 \times 10^{23} \, \text{molecules/mol}\), the total number of molecules per \(\text{cm}^2\) is \( \frac{20 \times 10^{-6} \times 6.022 \times 10^{23}}{892} \approx 1.35 \times 10^{16} \text{ molecules/cm}^2 \).

Calculate Photon Absorption Frequency

Find how often a single chlorophyll molecule absorbs a photon. Divide the total number of absorbed photons \(9.22 \times 10^{18} \text{ photons/min}\) by the total number of chlorophyll molecules \(1.35 \times 10^{16} \text{ molecules/cm}^2\) to find photons per chlorophyll molecule per minute: \(\frac{9.22 \times 10^{18}}{1.35 \times 10^{16}} \approx 683 \text{ photons/molecule/min}\). Therefore, each molecule absorbs a photon approximately every \( \frac{1}{683} \) minutes, or every \( \frac{60}{683} \approx 0.088\) seconds, or every 88 milliseconds.

Calculate Fraction of Excited Molecules

Finally, determine the fraction of chlorophyll molecules that are excited at any one time. Given the lifetime of an excited chlorophyll molecule is 1 ns (\(1 \times 10^{-9} \, \text{s}\)), during 1 second a molecule can absorb a photon every 0.088 seconds. Hence, the fraction excited is \(\frac{1 \times 10^{-9}}{0.088} \approx 1.14 \times 10^{-8}\).

Key concepts

Photosynthesis Energy

Photosynthesis is the process that plants use to convert light energy from the sun into chemical energy. This energy is stored in the bonds of glucose molecules, which serve as food for the plant. The efficiency of photosynthesis is crucial because it determines how much energy a plant can store.
Photosynthesis involves two main phases:

• The light-dependent reactions, which require sunlight and take place in the chloroplasts. These reactions capture light energy and store it temporarily in energy-rich molecules like ATP and NADPH.
• The light-independent reactions, or Calvin Cycle, which do not directly require light and occur in the stroma of the chloroplasts. Here, the stored energy from ATP and NADPH is used to convert carbon dioxide into glucose.

These processes together harness solar energy and contribute to the plant's growth and survival.

Chlorophyll Excitation

Chlorophyll, the green pigment in plants, plays a vital role in capturing light energy. It's responsible for the green coloration of leaves. When chlorophyll molecules absorb photons from sunlight, they become 'excited'. This state of excitement means that electrons in the chlorophyll molecule jump to a higher energy level.
This process is fundamental in the initial stages of the light-dependent reactions of photosynthesis. The excited electrons can then be transferred through a series of proteins embedded in the chloroplast membrane. This transfer helps to create a proton gradient, which drives the synthesis of ATP—the energy currency of the cell.

Photon Calculation

Photon calculation helps us understand how many photons are required for chlorophyll excitation and energy conversion. When sunlight strikes a leaf, only a fraction of its energy is absorbed by chlorophyll molecules.
The energy of a single photon can be calculated using the equation:

• \[ E = \frac{hc}{\lambda} \]

Where:

• \( E \) is the energy of the photon
• \( h \) is Planck's constant (\(6.63 \times 10^{-34} \text{ J}\cdot\text{s}\))
• \( c \) is the speed of light (\(3.00 \times 10^8 \text{ m/s}\))
• \(\lambda\) is the wavelength of light (~680 nm for chlorophyll absorption)

This helps us convert the total energy absorbed into the number of photons, which is critical to determining how often chlorophyll molecules are excited.

Chlorophyll Molecule

A chlorophyll molecule is a complex structure mainly composed of carbon, hydrogen, oxygen, nitrogen, and magnesium. It performs the essential task of absorbing light to initiate photosynthesis. Its structure allows it to capture light efficiently and transfer the absorbed energy for chemical reactions.
Each chlorophyll molecule can absorb a photon and transform its energy state. In the case of our exercise, chlorophyll concentration is measured, allowing us to calculate the number of these molecules within a given area. With this information, it's possible to determine the frequency of photon absorption per molecule, which provides insight into the photosynthetic efficiency of the leaf.
Understanding the behavior of chlorophyll at the molecular level is key to enhancing photosynthetic efficiency in crops, which could potentially boost food production globally.