Problem 51 - Chemistry An Atoms First Approach

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?

Step-by-Step Solution

Answer

The maximum wavelength of light that can provide enough energy to remove an electron from an iron atom is 275 nm.

Step 1: Write down the given values and the Planck-Einstein relation equation

We are given the energy E needed to remove an electron, which is \(7.21 \times 10^{-19} \mathrm{J}\). We'll also need the values for the Planck constant h (\(6.63 \times 10^{-34} \mathrm{Js}\)) and the speed of light c (\(3.00 \times 10^8 \frac{\mathrm{m}}{\mathrm{s}}\)). We want to find the wavelength λ. The Planck-Einstein relation is: \[E = \dfrac{hc}{\lambda}\]

Step 2: Rearrange the equation to solve for the wavelength λ

We want to find the value of λ. To do this, we need to rearrange the equation so that it is in the form λ =... . We can do this by multiplying both sides of the equation by λ and then dividing by E. This gives: \[\lambda = \dfrac{hc}{E}\]

Step 3: Substitute the known values and find the maximum wavelength

Now we can substitute the given values for h, c, and E into the equation: \[\lambda = \dfrac{(6.63 \times 10^{-34} \mathrm{Js})(3.00 \times 10^8 \frac{\mathrm{m}}{\mathrm{s}})}{7.21 \times 10^{-19} \mathrm{J}}\] Perform the calculation to find the value of λ: \[\lambda = \dfrac{(6.63 \times 10^{-34})(3.00 \times 10^8)}{7.21 \times 10^{-19}} \approx 2.75 \times 10^{-7} \mathrm{m}\] Since we usually express wavelength in nanometers (nm), we can convert the result by multiplying by \(10^9\): \[\lambda = 2.75 \times 10^{-7} \mathrm{m} \times 10^9 \frac{\mathrm{nm}}{\mathrm{m}} = 275 \mathrm{nm}\]

Step 4: State the final answer

The maximum wavelength of light that can provide enough energy to remove an electron from an iron atom is 275 nm.

Key Concepts

Planck-Einstein Relation

The Planck-Einstein relation forms a fundamental connection between the energy of a photon and its wavelength. This relationship is expressed by the equation:\[ E = \dfrac{hc}{\lambda} \]Where:

• \(E\) is the energy of the photon in joules (J).
• \(h\) is Planck’s constant, \(6.63 \times 10^{-34} \mathrm{Js}\).
• \(c\) is the speed of light, \(3.00 \times 10^8 \frac{\mathrm{m}}{\mathrm{s}}\).
• \(\lambda\) is the wavelength of the photon in meters (m).

This equation tells us that the energy of a photon is inversely proportional to its wavelength. That means shorter wavelengths have higher energy, and longer wavelengths have lower energy.
If you know energy (\(E\)), you can find the wavelength (\(\lambda\)) using this relation by rearranging the equation to:\[ \lambda = \dfrac{hc}{E} \]This formula helps in various applications, including calculating the properties of light required for specific tasks, like removing electrons from atoms.

Energy Quantization

Energy quantization refers to the idea that energy exists in discrete amounts, rather than a continuous range. This concept is pivotal in quantum mechanics and can be observed in different phenomena:- **Photon Energy**: Photons, the fundamental particles of light, carry energy in quantized packets called quanta.- **Quantized Energy Levels**: In atoms, electrons can occupy specific energy levels. They can transition between these levels by absorbing or emitting photons.The concept of quantized energy is crucial for understanding interactions at microscopic levels. Only photons with energy equivalent to the difference between energy levels can induce transitions.
This ties into how light of certain wavelengths can remove electrons, known as the photoelectric effect. Only light with energy equal to or greater than the energy holding the electron can free it. This principle underlies the calculation of the maximum wavelength of light that can remove an electron from an atom. The wavelength must correspond to the specific quantized energy needed for removal, in this case given by \(E\).

Electron Removal Energy

Electron removal energy, also known as ionization energy, is the energy necessary to remove an electron from an atom completely. This value tells us how strongly an electron is bound to an atom.In our exercise, the electron removal energy from an iron atom is given as \(7.21 \times 10^{-19} \mathrm{J}\). To remove the electron, light with energy meeting or exceeding this amount is required.

Calculating Maximum Wavelength

Light with a longer wavelength has less energy according to the Planck-Einstein relation. Therefore, to calculate the maximum wavelength capable of removing the electron, we use the following equation:\[ \lambda = \dfrac{hc}{E} \]Substituting known values for \(h\), \(c\), and \(E\) allows us to determine the maximum wavelength. The resulting value is 275 nm.
This particular wavelength represents the longest wavelength with sufficient energy to remove the electron. Understanding this helps in designing systems that might control or harness such interactions, like in spectroscopy or material science.