Question

A photon of ultraviolet (UV) light possesses enough energy to mutate a strand of human DNA. What is the energy of a single UV photon and a mole of UV photons having a wavelength of \(25 \mathrm{~nm} ?\)

Short answer

The energy of a single UV photon with a wavelength of \(25 nm\) is approximately \(7.97 × 10^{-19} J\), and the energy of a mole of UV photons is approximately \(4.80 × 10^5 J\).

Step-by-step solution

Convert the wavelength to meters

First, we need to convert the wavelength from nanometers (nm) to meters (m) since the other constants are in SI units. We know that 1 nm is equal to 1 × 10^-9 m. So, we can convert the given wavelength as follows: \( \lambda = 25 ~nm * \frac{1 × 10^{-9} ~m}{1 ~nm} = 25 × 10^{-9} ~m \)

Calculate the energy of a single photon

Now, we can plug in the values for Planck's constant (h), the speed of light (c), and the wavelength (λ) into the equation E = h * (c / λ) to find the energy of a single photon. Planck's constant (h) = 6.626 × 10^-34 Js Speed of light (c) = 3.00 × 10^8 m/s \( E = 6.626 × 10^{-34} ~Js * \frac{3.00 × 10^8 ~m/s}{25 × 10^{-9} ~m} \) Solve for E: \( E ≈ 7.97 × 10^{-19} ~J \) So, the energy of a single UV photon is approximately 7.97 × 10^-19 J.

Calculate the energy of a mole of photons

Now, we will find the energy of a mole of photons by multiplying the energy of a single photon by Avogadro's number (6.022 x 10^23). \( Energy~of~a~mole~of~photons = E × Avogadro's~number \) \( Energy~of~a~mole~of~photons = 7.97 × 10^{-19} ~J × 6.022 × 10^{23}~photon^{-1} \) Solve for the total energy: \( Energy~of~a~mole~of~photons ≈ 4.80 × 10^5~J \) So, the energy of a mole of UV photons is approximately 4.80 × 10^5 J.