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 that can remove an electron from an iron atom is \(2.76 \times 10^{-7}\) meters.

Step-by-step solution

Understand the relationship between energy and wavelength

The energy of a photon can be calculated using the equation: \( E = h\cdot f \) where E is the energy, h is Planck's constant (approximately \(6.63 \times 10^{-34} \mathrm{Js}\)), and f is the frequency of the photon. Since the frequency is related to the speed of light \( c \) by the equation and the wavelength \(\lambda\): \(f = c / \lambda \), We can rewrite the energy equation in terms of wavelength as follows: \( E = h\cdot c / \lambda \) We are given the energy required to remove an electron from an iron atom and asked to find the maximum wavelength of light that can do this, so we will solve the equation for the wavelength \(\lambda\):

Solve the energy equation for wavelength

Now we need to solve the equation for \(\lambda\). To isolate \(\lambda\), we will first divide both sides by \(h \cdot c\): \( \frac{E}{h\cdot c} = \frac{1}{\lambda} \) Now, we can take the reciprocal of both sides to find the value of \(\lambda\): \( \lambda = \frac{1}{\frac{E}{h\cdot c}} \)

Plug in the given energy value and constants

Using the given energy value, and the constants for Planck's constant and the speed of light, we can now calculate the wavelength: \( \lambda = \frac{1}{\frac{7.21 \times 10^{-19} \mathrm{J}}{(6.63 \times 10^{-34} \mathrm{Js}) \cdot (3.00 \times 10^{8} \, \mathrm{m/s})}} \)

Calculate the maximum wavelength

Now, we perform the calculations and find the maximum wavelength of light that can remove an electron from an iron atom: \( \lambda = \frac{1}{\frac{7.21 \times 10^{-19} \mathrm{J}}{(6.63 \times 10^{-34} \mathrm{Js}) \cdot (3.00 \times 10^{8} \, \mathrm{m/s})}} \) \( \lambda = \frac{1}{\frac{7.21 \times 10^{-19}}{1.989 \times 10^{-25}}} \) \( \lambda = \frac{1}{3.62 \times 10^6 \, \mathrm{m^{-1}}} \) \( \lambda = 2.76 \times 10^{-7} \, \mathrm{m} \) So, the maximum wavelength of light that can remove an electron from an iron atom is \(2.76 \times 10^{-7}\) meters.