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First, we need to convert the given dissociation energy from kJ/mol to Joules per photon. To do this, we will divide the given energy by Avogadro's number and convert kJ to Joules: Dissociation energy = 210 kJ/mol 1 kJ = 1000 Joules So, the dissociation energy is \(210 \times 1000 = 210,000\) Joules/mol Now, we need to divide this energy by Avogadro's number to get the energy per photon: Energy per photon = \(\frac{210,000\,\text{J/mol}}{6.022 \times 10^{23}\,\text{photons/mol}} = 3.488 \times 10^{-19}\) J/photon

Next, we will use Planck's constant, denoted by \(h\), to calculate the frequency of the photons. The formula to relate energy and frequency is given by: Energy = \(h \times \text{frequency}\) Where \(h = 6.626 \times 10^{-34}\) Js We can calculate the frequency by rearranging the formula: Frequency = \(\frac{\text{Energy}}{h}\) = \(\frac{3.488 \times 10^{-19}\,\text{J}}{6.626 \times 10^{-34}\,\text{Js}} = 5.266 \times 10^{14}\) Hz

Now that we have the frequency, we can use the speed of light, denoted by \(c\), to calculate the maximum wavelength of photons that can cause the dissociation. The formula to relate frequency and wavelength is given by: Speed of light = Frequency × Wavelength Where \(c = 3.0 \times 10^{8}\) m/s We can calculate the wavelength by rearranging the formula: Wavelength = \(\frac{c}{\text{Frequency}}\) = \(\frac{3.0 \times 10^{8}\,\text{m/s}}{5.266 \times 10^{14}\,\text{Hz}} = 5.700 \times 10^{-7}\) m Since the question asks for the wavelength in nanometers (nm), we can convert this value from meters to nanometers: Wavelength = \(5.700 \times 10^{-7}\,\text{m} \times \frac{10^{9}\,\text{nm}}{1\,\text{m}} = 570\) nm

Now that we have calculated the maximum wavelength of photons that can cause the dissociation of a carbon-bromine bond, we can present the final answer: The maximum wavelength of photons that can cause \(\mathrm{C}-\mathrm{Br}\) bond dissociation is \(570\,\text{nm}\).