Question

The lowest-frequency light that will produce the photoelectric effect is called the threshold frequency. (a) The threshold frequency for indium is \(9.96 \times\) \(10^{14} \mathrm{s}^{-1} .\) What is the energy, in joules, of a photon of this radiation? (b) Will indium display the photoelectric effect with UV light? With infrared light? Explain.

Short answer

The energy of a photon with this frequency is \(6.604 × 10^{-19} J\). Indium will show the photoelectric effect with UV light but not with infrared light.

Step-by-step solution

Calculate the Energy of a Photon

Using the given threshold frequency for indium, \(\(9.96 \times 10^{14} s^{-1}\)\) and Planck's constant, calculate the energy of a photon by applying the energy formula \(E = hν\). So, \(E = 6.626 \times 10^{-34} Js \times 9.96 \times 10^{14} s^{-1} = 6.604 × 10^{-19} J\).

Identification of the Effect of UV and Infrared Light

UV light has a frequency range of \(10^{16} to 10^{15} s^{-1}\) and infrared light has a frequency from \(10^{14} to 10^{12} s^{-1}\). As indium's threshold frequency is \(9.96 \times 10^{14} s^{-1}\), it is less than the frequency of UV light and greater than the frequency of infrared light. Hence, indium will show the photoelectric effect with UV light but not with infrared light, as the frequency of infrared light is less than the threshold frequency.

Key concepts

Threshold Frequency

The concept of threshold frequency is crucial when understanding the photoelectric effect. It refers to the minimum frequency of light needed to release electrons from a material. In other words, for light to cause electrons to be emitted from a surface, such as indium, its frequency must be at least as high as the threshold frequency of the material.

When light with a frequency below this threshold strikes a material, it will not cause electron emission, regardless of the light’s intensity. Thus, threshold frequency is a property inherent to specific materials, acting as a cut-off point for the photoelectric effect to take place.

For indium, the threshold frequency is given as \(9.96 \times 10^{14} \, \text{s}^{-1}\). This means any light with a frequency less than this value will not eject electrons, while light with a frequency equal to or greater will.

Indium

Indium is a metallic element used in various technological applications, including in semiconductors and solar cells. It has unique photoelectric properties that make it interesting for studies involving the photoelectric effect.

One key property is its threshold frequency, \(9.96 \times 10^{14} \, \text{s}^{-1}\), which is significant when considering its response to different types of electromagnetic radiation.

• Indium can emit electrons when exposed to light with sufficiently high frequencies.
• It does not respond to lower frequency light such as infrared, which is below its threshold.

Planck's Constant

Planck's constant is a fundamental constant in physics, denoted as \(h\), and it is crucial for calculating the energy of photons in the photoelectric effect. Its value is approximately \(6.626 \times 10^{-34} \, \text{Js}\).

When determining the energy of a photon, the formula \(E = hu\) is used, where \(E\) is energy, \(h\) is Planck's constant, and \(u\) (nu) represents frequency.

• For indium, using the threshold frequency \(9.96 \times 10^{14} \, \text{s}^{-1}\), the energy of a photon can be calculated.
• In this case, the photon energy equals \(6.604 \times 10^{-19}\) Joules.

This calculation forms a basis for understanding how different frequencies of light affect the emission of photoelectrons.

Ultraviolet Light

Ultraviolet (UV) light encompasses frequencies higher than visible light, often falling in the range from \(10^{15} \, \text{to} \, 10^{16} \, \text{s}^{-1}\). Because these frequencies are above indium's threshold frequency, UV light can cause electrons to be emitted from its surface.

UV light's ability to induce the photoelectric effect is why it is frequently used in scientific applications to study material properties and electron behavior.

• For metals like indium, UV light is highly effective in ejecting electrons.
• This makes it preferred in photoelectric studies and instruments.

Infrared Light

Infrared light has a frequency range generally lower than visible light, typically between \(10^{12} \, \text{to} \, 10^{14} \, \text{s}^{-1}\). Its frequencies fall below the threshold frequency of indium, which is \(9.96 \times 10^{14} \, \text{s}^{-1}\).

Because of this, infrared light cannot produce the photoelectric effect in indium.

• Infrared light lacks the necessary frequency to release electrons from materials such as indium.
• While useful in many applications, it won't trigger electron emission in photoelectric scenarios.

This highlights the importance of frequency in determining whether the photoelectric effect can occur.