Question

The work function of a certain material is \(5.8 \mathrm{eV}\). What is the photoelectric threshold for this material?

Short answer

Answer: The photoelectric threshold frequency for the material is approximately \(1.403 \times 10^{15} \mathrm{Hz}\).

Step-by-step solution

Express the work function as energy in Joules

To find the threshold frequency, we will first need to convert the given work function from electron volts (eV) to Joules (J). We can use the conversion factor, which is \(1 \mathrm{eV} = 1.602 \times 10^{-19} \mathrm{J}\).

Calculate the energy in Joules

Now, let's multiply the given work function (\(5.8 \mathrm{eV}\)) by the conversion factor to find the energy in Joules. Energy (J) = Work function (eV) × Conversion factor Energy (J) = \(5.8 \times 1.602 \times 10^{-19} \mathrm{J/eV}\)

Find the photoelectric threshold frequency

Using Planck's equation, we can find the photoelectric threshold frequency, where \(E = h\nu\). Here, \(E\) is the energy in Joules, \(h\) is the Planck's constant (\(6.626 \times 10^{-34} \mathrm{Js}\)), and \(\nu\) is the threshold frequency. We need to find the value of \(\nu\), so we can rearrange Planck's equation to solve for \(\nu\): \(\nu = \frac{E}{h}\) Now, we can substitute the values of \(E\) and \(h\) and compute the threshold frequency: \(\nu = \frac{5.8 \times 1.602 \times 10^{-19} \mathrm{J}}{6.626 \times 10^{-34} \mathrm{Js}}\)

Solve for the threshold frequency

After calculating the expression, we obtain the photoelectric threshold frequency for the material: \(\nu \approx 1.403 \times 10^{15} \mathrm{Hz}\) This is the minimum frequency of incident light that can release electrons from the material.

Key concepts

Work Function

The work function is a crucial concept in the photoelectric effect. It refers to the minimum energy needed to remove an electron from the surface of a metal. Think of it as a barrier that the energy of the incoming light must overcome to liberate an electron.

• Measured in electron volts (eV) or Joules (J).
• Uniquely characteristic to every material due to its atomic structure.

In the given problem, we have a work function of 5.8 eV. To work with it effectively in physics, especially when calculating the photoelectric threshold frequency, it is often necessary to convert this energy into Joules. The conversion is done using the factor 1 eV = 1.602 × 10^{-19} J.
This unit conversion helps when you later use this energy in formulas that require units in Joules, making calculations clearer and consistent.

Planck's Constant

Planck's constant plays an essential role in the quantum mechanical world. It's a fundamental constant that connects the energy of photons (particles of light) to their frequency; the relationship is linear. The equation is given by:\[ E = h u \]where \(E\) is energy measured in Joules, \(h\) is Planck's constant, and \(u\) is the frequency of the light.

• The value of Planck's constant is \(6.626 \times 10^{-34} \text{Js}\).
• This relationship is central to understanding how photons interact with electrons in the photoelectric effect.

In our photoelectric calculations, Planck's constant allows us to find the threshold frequency of photons needed to eject an electron when we know the work function. This conversion from energy (Joules) to frequency (Hz) is fundamental to analyzing photoelectric interactions.

Threshold Frequency

Threshold frequency is another key concept related to the photoelectric effect. It describes the minimum frequency of incident light required to eject electrons from a material's surface. If the frequency of the light is below this threshold, no electrons will be emitted regardless of the light's intensity.

• Calculated using the equation \(u = \frac{E}{h}\), where \(E\) is the energy (Joules) from the work function, and \(h\) is Planck's constant.
• A fundamental property of the material itself, unaffected by changes in light intensity.

From the original exercise, we determined this frequency to be approximately \(1.403 \times 10^{15} \text{Hz}\). This tells us that any light below this frequency won’t have enough energy to overcome the material's work function, thus failing to release electrons.