The bond energies in DNA are about 300 kJ/mol. Calculate the maximum wavelength, in nanometers, of a photon required to break one bond.
The correct answer and explanation is :
To calculate the maximum wavelength of a photon required to break one bond in DNA, we can use the relationship between the energy of a photon and its wavelength, which is given by the equation:
[
E = \frac{hc}{\lambda}
]
Where:
- ( E ) is the energy of the photon (in joules),
- ( h ) is Planck’s constant ((6.626 \times 10^{-34}) J·s),
- ( c ) is the speed of light ((3.00 \times 10^8) m/s),
- ( \lambda ) is the wavelength of the photon (in meters).
The energy is given as 300 kJ/mol. To calculate the energy of a single photon, we must first convert this energy into joules per photon.
Step 1: Convert the given energy from kJ/mol to joules per photon
Since 1 mole of DNA bonds contains Avogadro’s number (( N_A )) of molecules ((6.022 \times 10^{23}) molecules), we can calculate the energy per molecule.
The total energy in joules for 1 mole of DNA bonds is:
[
300 \, \text{kJ/mol} = 300 \times 10^3 \, \text{J/mol}
]
Now, the energy per bond (per photon) is:
[
E_{\text{photon}} = \frac{300 \times 10^3}{6.022 \times 10^{23}} \, \text{J}
]
[
E_{\text{photon}} \approx 4.98 \times 10^{-19} \, \text{J}
]
Step 2: Use the photon energy equation to solve for wavelength
We know that ( E = \frac{hc}{\lambda} ), so solving for ( \lambda ) (wavelength), we get:
[
\lambda = \frac{hc}{E}
]
Substitute the known values:
[
\lambda = \frac{(6.626 \times 10^{-34} \, \text{J·s}) (3.00 \times 10^8 \, \text{m/s})}{4.98 \times 10^{-19} \, \text{J}}
]
[
\lambda \approx 4.00 \times 10^{-7} \, \text{m}
]
Step 3: Convert the wavelength into nanometers
Since 1 meter = (10^9) nanometers:
[
\lambda \approx 4.00 \times 10^{-7} \, \text{m} = 400 \, \text{nm}
]
Final Answer:
The maximum wavelength of a photon required to break one bond in DNA is 400 nm.
Explanation:
This calculation shows how the energy required to break a bond in DNA (300 kJ/mol) relates to the maximum wavelength of light needed to break that bond. By converting the energy into joules and using the equation that relates energy and wavelength for photons, we determined that a photon with a wavelength of 400 nm would have enough energy to break the bond. This wavelength corresponds to ultraviolet light, which is known to be capable of breaking chemical bonds, such as those in DNA.