Question

Applying Models In discussions of the photoelectric effect, the minimum energy needed to remove an electron from the metal is called the threshold energy and is a characteristic of the metal. For example, chromium, Cr, will emit electrons when the wavelength of the radiation is 284 nm or less. Calculate the threshold energy for chromium. (Hint: You will need to use the two equations that describe the relation- ships between wavelength, frequency, speed of light, and Planck’s constant.)

Short answer

The threshold energy for chromium is \ 7.00 \times 10^{-19} \text{J} \.

Step-by-step solution

Identify the Given Information

The wavelength of the radiation for chromium, \(\lambda=284 \text{nm}\). Convert this to meters by using the conversion \(1 \text{nm} = 10^{-9} \text{m} \Rightarrow \lambda = 284 \text{nm} = 284 \times 10^{-9} \text{m} = 2.84 \times 10^{-7} \text{m} \).

Recall Relevant Equations

Two equations are relevant here: \(c = \lambda \u \) (relationship between wavelength, frequency, and speed of light) and \(E = h \u \) (relationship between energy, Planck’s constant, and frequency).

Calculate the Frequency

Use \ c = \lambda \u \ to calculate the frequency \( \u\): \ \(c = 3.00 \times 10^{8} \text{m/s}\), \(\lambda = 2.84 \times 10^{-7} \text{m}\). Thus, \( \u = \frac{c}{\lambda} = \frac{3.00 \times 10^{8}}{2.84 \times 10^{-7}} \text{s}^{-1} = 1.056 \times 10^{15} \text{s}^{-1} \).

Calculate the Threshold Energy

Use \ E = h \u \ to find the threshold energy \ E \: Planck’s constant \(h = 6.626 \times 10^{-34} \text{Js} \) and the frequency \ \(u = 1.056 \times 10^{15} \text{s}^{-1} \). Thus, \( E = h \u = 6.626 \times 10^{-34} \times 1.056 \times 10^{15} = 7.00 \times 10^{-19} \text{J} \. \)

Key concepts

Threshold Energy

Threshold energy, often referred to as the work function, is the minimum energy required to remove an electron from the surface of a metal.
This concept is critical when understanding the photoelectric effect, which is a phenomenon where electrons are emitted from a metal when it absorbs light. For different metals, this threshold energy varies because it depends on the metal's atomic structure.
The importance of threshold energy is highlighted in the photoelectric effect. When light strikes a metal surface, electrons are emitted only if the light's energy is greater than the metal's threshold energy.
If the energy of the incident light is less than the threshold energy, no electrons will be emitted regardless of the light's intensity.
This shows that the photoelectric effect is centered on the energy of photons rather than their number.

Planck’s Constant

Planck’s constant is a fundamental constant in physics, symbolized as \( h \).
It links the energy of a photon to its frequency and plays a crucial role in quantum mechanics.
Mathematically, this is expressed as \( E = h u \), where \( E \) is energy, \( h \) is Planck’s constant \( (6.626 \times 10^{-34} \text{Js}) \), and \( u \) is the frequency.
Planck’s constant is significant because it quantifies the discrete energy levels in atomic and subatomic systems.
In the context of the photoelectric effect, Planck's constant is utilized to calculate the energy of photons, thereby determining whether they can eject electrons from a metal surface.

Wavelength and Frequency Relationship

Understanding the relationship between wavelength and frequency is key in solving problems related to the photoelectric effect.
The wavelength of light is the distance between successive peaks of a wave, while frequency is the number of peaks that pass a point in one second.
They are inversely related and are linked by the speed of light formula: \( c = u \times \text{ wavelength} \).
This means that an increase in frequency results in a decrease in wavelength and vice versa.
The speed of light \( (c) \) is a constant \( 3.00 \times 10^{8} \text{m/s} \).
For instance, if we know the wavelength of light, we can use this formula to find its frequency, which is crucial for calculating the energy of the photons using Planck's equation.

Speed of Light

The speed of light in a vacuum, denoted by \( c \), is one of the most important constants in physics.
Its value is approximately \( 3.00 \times 10^{8} \text{m/s} \).
This constant shows how fast light waves travel in a vacuum and is key to connecting the frequency and wavelength of electromagnetic waves.
Understanding the speed of light is essential when dealing with the photoelectric effect.
The relationship \( c = u \times \text{wavelength} \) lets us determine either the frequency or wavelength when one is known.
This allows us to work backwards from known values, such as wavelength, to find the frequency and, ultimately, the energy of the light photons impacting a material.