Problem 114 - Chemistry The Molecular Science

Question

A certain minimum energy, \(E_{\text {min }}\) is required to eject an electron from a photosensitive surface. Any energy absorbed beyond this minimum gives kinctic cnergy to the cjected clectron. When \(540 .-\mathrm{nm}\) light falls on a cesium surface, an electron is ejected with a kinctic energy of \(6.69 \times 10^{-20} \mathrm{~J}\). When the wavelength is \(400 \mathrm{nm}\), the kinctic cnergy is \(1.96 \times 10^{-19} \mathrm{~J}\). (a) Calculate \(E_{\text {min }}\) for cesium, in joules. (b) Calculate the longest wavelength, in nanometers, that will eject an electron from cesium.

Step-by-Step Solution

Answer

a) \(E_{\text{min}} = 3.01 \times 10^{-19} \text{J}\), b) Longest wavelength = 660 nm.

Step 1: Understand the problem

We need to find the minimum energy required to eject an electron, denoted as \(E_{\text{min}}\), from a photosensitive surface using the photoelectric effect. We also need to determine the longest wavelength able to eject an electron.

Step 2: Use the Photoelectric Equation

According to the photoelectric effect, the energy of light \(E\) is given by:\[ E = E_{\text{min}} + KE \]where \(E_{\text{min}}\) is the minimum energy (work function), and \(KE\) is the kinetic energy of the ejected electron. The energy of the light can also be calculated using:\[ E = \frac{hc}{\lambda} \]where \(h\) is Planck’s constant \((6.626 \times 10^{-34}\, \text{Js})\), \(c\) is the speed of light \((3.00 \times 10^8\, \text{m/s})\), and \(\lambda\) is the wavelength of the light.

Step 3: Calculate Energy for 540 nm

First, we calculate the energy of a photon with 540 nm wavelength:\[ E_{540} = \frac{6.626 \times 10^{-34} \cdot 3.00 \times 10^8}{540 \times 10^{-9}} \approx 3.68 \times 10^{-19}\, \text{J} \]Using this, find \(E_{\text{min}}\) using the kinetic energy provided:\[ E_{\text{min}} = E_{540} - 6.69 \times 10^{-20} \approx 3.68 \times 10^{-19} - 6.69 \times 10^{-20} \]\[ E_{\text{min}} \approx 3.01 \times 10^{-19} \text{J} \]

Step 4: Calculate Energy for 400 nm

Calculate the energy of a photon with a 400 nm wavelength:\[ E_{400} = \frac{6.626 \times 10^{-34} \cdot 3.00 \times 10^8}{400 \times 10^{-9}} \approx 4.97 \times 10^{-19}\, \text{J} \]Check this calculation against \(E_{\text{min}}\):\[ E_{\text{min}} = E_{400} - 1.96 \times 10^{-19} \approx 4.97 \times 10^{-19} - 1.96 \times 10^{-19} \]\[ E_{\text{min}} \approx 3.01 \times 10^{-19} \text{J} \]This confirms our previous \(E_{\text{min}}\) calculation.

Step 5: Calculate the Longest Wavelength

Using \(E = E_{\text{min}}\) to find the threshold wavelength, solve for \(\lambda\):\[ \lambda_{\text{max}} = \frac{hc}{E_{\text{min}}} = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{3.01 \times 10^{-19}} \]\[ \lambda_{\text{max}} \approx 660 \times 10^{-9}\, \text{m} = 660\, \text{nm} \]

Key Concepts

Work Function

The work function is a core concept in understanding the photoelectric effect. It is the minimum amount of energy required to remove an electron from the surface of a material. For each material, this work function is unique. In the context of the photoelectric effect, it represents the threshold energy that incoming photons must have to eject electrons from the material's surface. Any energy absorbed beyond this work function provides kinetic energy to the ejected electrons.

For cesium, in our exercise, we calculated the work function, denoted as \(E_{\text{min}}\), to be \(3.01 \times 10^{-19} \text{J}\). This value was derived using the energies of electrons ejected by photons with known wavelengths.

Kinetic Energy of Electrons

Kinetic energy of electrons is another crucial concept in the photoelectric effect. After a photon transfers its energy to an electron, any energy that exceeds the work function is transformed into kinetic energy for the electron. The formula used in this context is:

• \( KE = E - E_{\text{min}} \)

Here, \(E\) represents the total energy of the incoming photon, and \(E_{\text{min}}\) is the work function. In the given exercise, the kinetic energy of the electrons was calculated when different light wavelengths hit the cesium surface.

For 540 nm light, the kinetic energy was \(6.69 \times 10^{-20} \text{J}\), and for 400 nm light, it was \(1.96 \times 10^{-19} \text{J}\). These kinetic energies were useful for confirming the consistency of the calculated work function.

Photon Energy Calculation

The energy of a photon is pivotal in understanding its ability to eject electrons in the photoelectric effect. Photon energy can be calculated using the equation:

• \(E = \frac{hc}{\lambda}\)

In this formula, \(h\) is Planck's constant \((6.626 \times 10^{-34} \text{Js})\), \(c\) is the speed of light \((3.00 \times 10^8 \text{m/s})\), and \(\lambda\) is the wavelength of the light.
Using the given examples, for a wavelength of 540 nm, the photon energy was calculated to be approximately \(3.68 \times 10^{-19} \text{J}\). For a wavelength of 400 nm, it was \(4.97 \times 10^{-19} \text{J}\). These calculations directly contribute to the understanding of electron ejection and subsequent kinetic energy calculations.