Question

It takes \(7.21 \times 10^{-19} \mathrm{~J}\) of energy to remove an electron from an iron atom. What is the maximum wavelength of light that can do this?

Short answer

The maximum wavelength of light capable of removing an electron from the iron atom is approximately \(275.6 \mathrm{~nm}\).

Step-by-step solution

Identify the relevant equations and constants

For this problem, we need the following equation: Planck's equation: \(E = h \times f\) where: - \(E\) is the energy of the light (in Joules), - \(h\) is the Planck's constant (\(6.626 \times 10^{-34} \mathrm{~J \cdot s}\)), - \(f\) is the frequency of the light (in Hertz). Also, we need the relation between frequency and wavelength of light: \(c = \lambda \times f\) where: - \(c\) is the speed of light (\(3.0 \times 10^8 \mathrm{~m/s}\)), - \(\lambda\) is the wavelength of the light (in meters).

Express the frequency in terms of energy

Now we will solve the Planck's equation for the frequency: \(\Rightarrow f = \frac{E}{h}\)

Replace the energy value in the equation

Now replace the given energy value in the equation: \(f = \frac{7.21 \times 10^{-19} \mathrm{~J}}{6.626 \times 10^{-34} \mathrm{~J \cdot s}}\)

Calculate the frequency

Now calculate the frequency, giving: \(f \approx 1.088 \times 10^{15} \mathrm{~Hz}\)

Relate frequency and wavelength

Now we will use the relation between frequency and wavelength: \(\lambda = \frac{c}{f}\)

Compute the maximum wavelength

Now substitute the frequency value and speed of light into the equation to find the maximum wavelength: \(\lambda = \frac{3.0 \times 10^8 \mathrm{~m/s}}{1.088 \times 10^{15} \mathrm{~Hz}}\) \(\lambda \approx 2.756 \times 10^{-7} \mathrm{~m}\) Since the wavelength is often expressed in nanometers, we can convert the result: \(\lambda = 2.756 \times 10^{-7} \mathrm{~m} \times \frac{10^9 \mathrm{~nm}}{1\mathrm{~m}} \approx 275.6 \mathrm{~nm}\) Therefore, the maximum wavelength of light capable of removing an electron from the iron atom is approximately \(275.6 \mathrm{~nm}\).