Ultraviolet radiation and radiation of shorter wavelengths can damage biological molecules because they carry enough energy to break bonds within the molecules. a carbon-carbon bond requires 348 kj/mol to break.

what is the longest wavelength of radiation with enough energy to break carbon-carbon bonds?

Longest wavelength = 343.7 nm

In this question we need to first use the concept of energy of a photon.

Energy of a photon, E, is given by the formula, E = hf, where h is the plank's constant, f is the frequency.

But since, f is given by dividing speed, c, by wavelength, λ, then;

E = hc/λ

We are given 348 kJ/mol required to break carbon-carbon bonds.

We know that; 1 mole of bonds = 6.022 × 10^23 bonds.

We are required to find the longest wavelength with enough energy to break the C-C bonds.

This can be worked out in simple steps:

Step 1:  Energy required to break one bond (kJ/bond)

1 mole of bonds = 6.022 × 10^23 bonds.

Therefore;

348 kJ = 6.022 × 10^23 bonds.

Thus;

1 bond = 348 kJ ÷ 6.022 × 10^23 bonds.

           =  5.778 x 10^-22 kJ

But; 1000 joules = 1 kJ

Hence; energy per bond =  5.778 x 10^-19 Joules

Step 2: Energy per photon

Breaking one bond requires energy equivalent to energy of a photon.

Therefore;

1 photon = 5.778 x 10^-19 Joules

              = 5.778 x 10^-19 J/photon

Step 3: Calculating the wavelength

From the equation of energy of a photon;

E = hc/λ

h is the plank's constant = 6.626 × 10^-34 J/s

c is the speed of light in vacuum = 2.9998 × 10^8 m/s

E is the energy of a photon =  5.778 x 10^-19 Joules

Therefore, making λ (wavelength) the subject;

       = 3.437 x 10^-7 m

But; 1 nm = 10^-9 m

Thus;

wavelength = 343.7 nm

Therefore, the longest wavelength of the radiation will be 343.7 nm