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Chapter 7: Problem 42

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

Expert verified
The energy of a single UV photon with a wavelength of 25 nm is approximately \(7.95 \times 10^{-18}\) Joules, and a mole of such UV photons has an energy of approximately \(4.79 \times 10^6\) Joules per mole.

Step by step solution

01

Calculate the frequency of the UV photon

We are given the wavelength of the photon (\(\lambda\)), so we can calculate the frequency (\(\nu\)) by using the formula: \(\nu = \frac{c}{\lambda}\) \(\nu = \frac{3 \times 10^8 m/s}{25 \times 10^{-9} m} = 12 \times 10^{15} Hz\) The frequency of the UV photon is \(12 \times 10^{15}\) Hz.
02

Calculate the energy of a single UV photon

Now that we have the frequency of the photon, we can find the energy of the single photon using the Planck's equation: \(E = h\times\nu\) Planck's constant \(h = 6.626 \times 10^{-34} Js\) \(E = 6.626 \times 10^{-34}Js \times (12 \times 10^{15} Hz) = 7.95 \times 10^{-18} J\) The energy of a single UV photon is approximately \(7.95 \times 10^{-18}\) Joules.
03

Calculate the energy of a mole of UV photons

Now, we will find the energy of a mole of photons by multiplying the energy of a single photon by the Avogadro's number (N): Energy of mole of UV photons = (\(7.95 \times 10^{-18} J\)) * (\(6.022 \times 10^{23}\) \(mol^{-1}\)) Energy of mole of UV photons = \(4.79 \times 10^6 J/mol\) The energy of a mole of UV photons is approximately \(4.79 \times 10^6\) Joules per mole. In summary, a single UV photon has an energy of \(7.95 \times 10^{-18}\) Joules, and a mole of UV photons has an energy of \(4.79 \times 10^6\) Joules per mole.

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