Answer
$386\text{ nm}$
Work Step by Step
To determine the maximum wavelength of light that can cause the given reaction, we need to use the energy change for the reaction and the relationship between energy and wavelength.
The energy change for the reaction is given as \(3.10 \times 10^{2} \, \mathrm{kJ/mol}\). This energy change corresponds to the energy of one mole of photons, as each photon carries a specific amount of energy.
The relationship between energy and wavelength is given by the equation:
\[E = \dfrac{hc}{\lambda}\]
where \(E\) is the energy of a photon, \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \mathrm{J \cdot s}\)), \(c\) is the speed of light (\(3.00 \times 10^{8} \, \mathrm{m/s}\)), and \(\lambda\) is the wavelength of light.
To find the maximum wavelength, we need to find the minimum energy required for the reaction. This occurs when all the energy is supplied by a single photon. Therefore, we can rearrange the equation to solve for the maximum wavelength:
\[\lambda = \dfrac{hc}{E}\]
Substituting the given values:
\[\lambda = \dfrac{(6.626 \times 10^{-34} \, \mathrm{J \cdot s})(3.00 \times 10^{8} \, )}{3.10 \times 10^{2}\times \frac{1}{6.022\times 10^{23}} \, }\]
\[\lambda \approx 3.86\times 10^{-7}\text{ m}=386\text{ nm}\]
Therefore, the maximum wavelength of light that can cause this reaction is \(386 \, \mathrm{nm}\).
