Problem 24 - Chemistry The Molecular Science

Question

A photoemissive material has a threshold energy, \(E_{\min }=\) \(5 \times 10^{-19} \mathrm{~J}\). Will \(300 . \mathrm{nm}\) radiation eject electrons from the material? Explain.

Step-by-Step Solution

Answer

Yes, 300 nm radiation will eject electrons since its photon energy exceeds the threshold energy.

Step 1: Identify Wavelength and Energy Relationship

To determine if electron ejection will occur, we need the energy of the incoming photon. This can be calculated from its wavelength using the formula: \[ E = \frac{hc}{\lambda} \] where \( h = 6.626 \times 10^{-34} \) J.s (Planck's constant), \( c = 3 \times 10^8 \) m/s (speed of light), and \( \lambda = 300 \times 10^{-9} \) m (wavelength of radiation).

Step 2: Calculate Energy of Incoming Photon

Substitute the known values into the formula: \[ E = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{300 \times 10^{-9}} \] Calculate:\[ E = \frac{1.9878 \times 10^{-25}}{300 \times 10^{-9}} = 6.626 \times 10^{-19} \text{ J} \]

Step 3: Compare Photon Energy with Threshold Energy

The calculated photon energy \( E = 6.626 \times 10^{-19} \text{ J} \) is compared to the threshold energy \( E_{\min} = 5 \times 10^{-19} \text{ J} \). Since \( E > E_{\min} \), the photon energy exceeds the threshold energy.

Step 4: Conclude on Electron Ejection

Since the photon energy \( (6.626 \times 10^{-19} \text{ J}) \) is greater than the threshold energy \( (5 \times 10^{-19} \text{ J}) \), electrons will be ejected from the material.

Key Concepts

Photon Energy Calculation

When discussing photoemission, understanding how to calculate photon energy is crucial. Photon energy is the amount of energy carried by a single photon, and it can be determined using its wavelength. The key formula used to calculate the energy of a photon is:\[ E = \frac{hc}{\lambda} \]Here:

• \(E\) represents the energy of the photon in joules (J).
• \(h\) is Planck's constant, a fundamental constant with the value \(6.626 \times 10^{-34} \text{ J.s}\).
• \(c\) is the speed of light in a vacuum, approximately \(3 \times 10^8 \text{ m/s}\).
• \(\lambda\) is the wavelength of the photon in meters (m).

This formula shows the inverse relationship between a photon's wavelength and its energy. A shorter wavelength implies a higher energy for the photon. It is important to apply this formula correctly for accurate calculations in problems involving photoemission.

Threshold Energy

The concept of threshold energy is significant in the field of photoemission. Threshold energy refers to the minimum energy required to liberate an electron from a material. Without sufficient energy, incident photons will not be able to eject electrons, no matter their quantity or intensity.In the context of the given exercise:

• The threshold energy, \(E_{\min}\), is given as \(5 \times 10^{-19} \text{ J}\).
• It serves as the benchmark photon energy must exceed for electrons to be emitted from a specific material.

If the energy of the incoming photon is less than threshold energy, photoemission will not occur. Thus, accurately determining and comparing photon energy to threshold energy is an essential step in problems involving ejection of electrons.

Wavelength-Energy Relationship

The wavelength-energy relationship is foundational in topics dealing with electromagnetic radiation and photoemission. The relationship indicates that the energy of a photon is inversely proportional to its wavelength. This means:

• A shorter wavelength results in a higher energy photon.
• A longer wavelength means a lower energy photon.

Understanding this relationship allows one to determine whether certain wavelengths can trigger phenomena like electron ejection from materials.Given the formula for photon energy:\[ E = \frac{hc}{\lambda} \]You see that as \(\lambda\) decreases, \(E\) increases. This principle is why ultraviolet light (short wavelengths) can cause chemical reactions or skin damage, whereas infrared light (longer wavelengths) typically cannot.In practical applications, knowing this relationship helps to predict if a certain material will exhibit photoemission when exposed to specific wavelengths of light.