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 approximately 275 nm. This is found by using the photon energy formula and Planck's constant to relate the given energy to the frequency and wavelength of the light.

Step-by-step solution

Understand the Photon Energy Formula

The photon energy formula relates the energy of a photon, E, with its frequency, ν, using Planck's constant, h: \(E = h\nu\) The wavelength of light is also related to the frequency by the speed of light constant, which is c: \(\nu = \frac{c}{\lambda}\) Our goal is to find the maximum wavelength \(\lambda\), so we will combine these equations and solve for \(\lambda\).

Combine Equations

Replace \(\nu\) in the photon energy equation with speed of light equation, so we get: \(E = h\frac{c}{\lambda}\) Now, solve for \(\lambda\): \(\lambda = \frac{hc}{E}\)

Insert Constants and Given Values

Now we insert the values for Planck's constant (h), the speed of light (c), and the provided energy (E) into the equation: h = \(6.63 \times 10^{-34} \mathrm{~J \cdot s}\) (Planck's constant) c = \(3.00 \times 10^8 \mathrm{~m/s}\) (speed of light) E = \(7.21 \times 10^{-19} \mathrm{~J}\) (given energy) \(\lambda = \frac{(6.63 \times 10^{-34}\mathrm{~J \cdot s}) \times (3.00 \times 10^8\mathrm{~m/s})}{(7.21 \times 10^{-19}\mathrm{~J})}\)

Calculate the Maximum Wavelength

Finally, we calculate the maximum wavelength (λ): \(\lambda \approx 2.75 \times 10^{-7} \mathrm{~m}\) To express the answer in nanometers, we have: \(\lambda \approx 275 \mathrm{~nm}\)

Conclusion

The maximum wavelength of light that can remove an electron from an iron atom is approximately 275 nm.

Key concepts

Photon Energy Formula

Understanding the relationship between energy and photons is crucial when studying photoelectric effects, where light frees electrons from materials. The photon energy formula is expressed as:

\(E = hu\),

where \(E\) represents the energy of a photon, \(u\) is the frequency of the light, and \(h\) is Planck's constant. When photons hit matter, they can transfer their energy to electrons, and if this energy is sufficient, it can overcome the work function, the energy needed to free electrons. This is showcased in exercises that require determining the release of electrons, triggered by certain wavelengths of light.

Planck's Constant

Planck's constant is a fundamental value in quantum physics, signifying the proportionality constant between the energy of a photon and the frequency of its associated electromagnetic wave. Represented by \(h\), the constant is:

\(h = 6.63 \times 10^{-34} \mathrm{J \cdot s}\),

making it an indispensable part of calculations dealing with quantum scales. Planck's constant connects the macroscopic world we see every day to the mysterious quantum realm, and is used widely to analyze phenomena such as the photoelectric effect, blackbody radiation, and quantum tunneling.

Speed of Light

The speed of light, denoted by \(c\), is a constant that light and all electromagnetic waves travel at in a vacuum. It is a foundation for our understanding of the universe and is precisely:

\(c = 3.00 \times 10^8 \mathrm{m/s}\),

which is unimaginably fast from a human perspective. In exercises involving the photoelectric effect, the speed of light relates to the frequency and wavelength of light, allowing us to dissect how changes in light's properties affect the energy of photons.

Wavelength of Light

Wavelength is a key concept in understanding light's behaviors and properties. It is the distance between successive peaks of a wave and is usually denoted by \(\lambda\) (lambda). Wavelength is inversely proportional to frequency, meaning the higher the frequency, the shorter the wavelength and vice-versa. This relationship is specified as:

\(\lambda = \frac{c}{u}\),

where \(c\) is the speed of light. In the scenario of the photoelectric effect, finding a maximum wavelength that can release an electron involves utilizing the known values of a material's work function, Planck's constant, and the speed of light to solve for \(\lambda\), illuminating which part of the electromagnetic spectrum has the necessary energy.