Problem 118 - Chemistry

Question

A photoelectric experiment was performed by separately shining a laser at \(450 \mathrm{nm}\) (blue light) and a laser at \(560 \mathrm{nm}\) (yellow light) on a clean metal surface and measuring the number and kinetic energy of the ejected electrons. Which light would generate more electrons? Which light would eject electrons with greater kinetic energy? Assume that the same amount of energy is delivered to the metal surface by each laser and that the frequencies of the laser lights exceed the threshold frequency.

Step-by-Step Solution

Answer

More electrons would be generated by the yellow light, however, the blue light would eject electrons with greater kinetic energy.

Step 1: Understand the Photoelectric Effect and Einstein's Equation

In the photoelectric effect, the energy of the incident light is absorbed by electrons, which in turn get ejected from the surface of a material. Albert Einstein formalized this observation through his photoelectric equation: \( E = hf - \Phi \), where E represents the kinetic energy of the ejected electrons, f is the frequency of the incident light, h is Planck's constant, and \(\Phi\) is the work function of the material (the minimum energy required to remove an electron from the surface). However, in this exercise it is stated that both lights exceed the threshold frequency, effectively making the work function negligible. The simplified equation to look at is then \( E = hf \).

Step 2: Apply the Properties of Light

Now it's important to understand the properties of light, specifically that the frequency and wavelength of light have an inverse relationship - as the frequency of light increases, its wavelength decreases and vice versa (assuming the speed of light, c, to be a constant). Therefore, blue light, which has a shorter wavelength (450 nm) than yellow light (560 nm), has a greater frequency.

Step 3: Determine the Number and Energy of Ejected Electrons

Since the exercise states that both lasers deliver the same amount of energy to the metal surface, the number of electrons ejected (i.e., the current observed) depends on the energy of the incident photons, which is given by \( E = hf \). Based on this, because blue light has a higher frequency than yellow light, each photon of blue light has more energy compared to each photon of yellow light. Therefore, given a set amount of energy, yellow light (which has lower energy per photon) would eject more electrons than blue light (which has higher energy per photon). However, since each blue light photon has more energy, any electron that absorbs a photon of blue light will be ejected with a greater kinetic energy compared to an electron that absorbs a photon of yellow light.

Key Concepts

Einstein's Photoelectric Equation

Einstein's photoelectric equation is a fundamental piece of our understanding of the photoelectric effect. It describes how light can cause the emission of electrons from a metal surface. This phenomenon is pivotal in understanding quantum mechanics and played a key role in the development of modern physics.
In simple terms, the equation is represented as \( E = hf - \Phi \). Here, \( E \) is the kinetic energy of ejected electrons, \( h \) is Planck's constant, \( f \) is the frequency of the incident light, and \( \Phi \) is the work function specific to each material.
When light hits the surface of a metal, its energy is transferred to the electrons. If this energy surpasses the work function, electrons are ejected from the surface. In this exercise, since the frequencies of the lasers exceed the threshold, the work function \( \Phi \) can be disregarded, simplifying to \( E = hf \). This equation indicates that the energy transferred to the electrons depends directly on the light frequency.

Frequency and Wavelength Relationship

One of the essential insights from the photoelectric effect relates to the relationship between frequency and wavelength of light. These two properties are inversely related, as represented by the equation \( c = \lambda f \), where \( c \) is the constant speed of light, \( \lambda \) is the wavelength, and \( f \) is the frequency of the light.
In practical terms, this means that light with a high frequency will have a shorter wavelength, and light with a low frequency will have a longer wavelength. For example, blue light with a wavelength of 450 nm has a higher frequency compared to yellow light of 560 nm. Consequently, blue light photons possess more energy compared to yellow light photons.
This relationship is crucial when analyzing the photoelectric effect as the frequency directly influences the energy carried by photons, thus impacting the emission and kinetic energy of electrons ejected from a metal surface.

Kinetic Energy of Ejected Electrons

The kinetic energy of the electrons ejected during the photoelectric effect provides pivotal insights into the behavior of particles at the quantum level. According to Einstein's photoelectric equation, \( E = hf - \Phi \), the kinetic energy of an ejected electron is directly proportional to the frequency of the incident light.
In scenarios where the frequency exceeds the threshold, the work function \( \Phi \) becomes negligible, simplifying the equation to \( E = hf \). This indicates that higher frequency (and thus, higher energy) light, like blue light, will result in electrons being ejected with greater kinetic energy.
However, it's important to note that the number of electrons ejected will vary inversely, as higher frequency light means fewer photons for a given amount of light energy. Therefore, while each electron ejected by blue light in the exercise will have more kinetic energy, yellow light will eject more electrons given the same energy input.