Question

When ultraviolet light with a wavelength of 400.0 nm falls on a certain metal surface, the maximum kinetic energy of the emitted photoelectrons is measured to be 1.10 eV. What is the maximum kinetic energy of the photoelectrons when light of wavelength 300.0 nm falls on the same surface?

Short answer

The maximum kinetic energy for 300.0 nm light is 2.13 eV.

Step-by-step solution

Understand the Problem

We need to find the maximum kinetic energy of photoelectrons emitted when light of a different wavelength (300.0 nm) falls on the same metal surface. This requires understanding the photoelectric effect where energy provided by light ejects electrons from a metal surface.

Use the Photoelectric Equation

The photoelectric equation is given by \( KE_{max} = hf - \phi \), where \( KE_{max} \) is the maximum kinetic energy of the photoelectrons, \( hf \) is the energy of the incident photons, and \( \phi \) is the work function of the metal.

Calculate Energy for Initial Wavelength

Convert the wavelength from nanometers to meters, \( 400.0 \text{ nm} = 400.0 \times 10^{-9} \text{ m} \). Calculate the initial photon energy using \( E = \frac{hc}{\lambda} \), where \( h = 6.626 \times 10^{-34} \text{ J·s} \), \( c = 3.00 \times 10^8 \text{ m/s} \), and \( \lambda = 400.0 \times 10^{-9} \text{ m} \).

Calculate Work Function

Since the maximum kinetic energy for \( 400.0 \text{ nm} \) is given as \( 1.10 \text{ eV} \), and knowing the energy in \( eV \), calculate \( \phi \) using the initial condition: \( 1.10 \text{ eV} = hf_{400} - \phi \). Solve for \( \phi \).

Calculate Energy for New Wavelength

Convert the new wavelength 300.0 nm to meters, \( 300.0 \text{ nm} = 300.0 \times 10^{-9} \text{ m} \). Calculate the photon energy using \( E_{300} = \frac{hc}{300.0 \times 10^{-9}} \), and convert from Joules to electron volts.

Calculate New Maximum Kinetic Energy

Use \( KE_{max,new} = E_{300} - \phi \) to find the new maximum kinetic energy in electron volts. Use the work function \( \phi \) calculated in Step 4 and the new photon energy from Step 5.

Key concepts

maximum kinetic energy

In the context of the photoelectric effect, **maximum kinetic energy** refers to the highest amount of kinetic energy that ejected photoelectrons can have once they leave the surface of a metal due to incident light. This maximum energy is denoted as \( KE_{max} \) and is dictated by the energy of the incoming photons minus the energy needed to liberate the electrons. - The kinetic energy of these electrons can be calculated with the photoelectric equation: \( KE_{max} = hf - \phi \), where: - \( hf \) is the energy of the incoming photons. - \( \phi \) is the work function, representing the minimum energy required to eject an electron from the metal surface.When different wavelengths of light are used, such as 400.0 nm and 300.0 nm, they are associated with different energies. The photon with the shorter wavelength (300.0 nm) has higher energy compared to the one with a longer wavelength (400.0 nm). Thus, the maximum kinetic energy of the emitted photoelectrons is higher with 300.0 nm light than with 400.0 nm light. Solving these values involves calculating the photon's energy at each wavelength and subtracting the work function, which remains constant for the metal in question.

work function

The **work function** is a crucial concept in the photoelectric effect. It represents the minimum energy required to remove an electron from the surface of a metal and is often denoted by the symbol \( \phi \). This energy barrier must be overcome for photoelectrons to be emitted from a metal surface when exposed to light. - The work function is akin to a threshold energy: - If the energy of the incoming photons is less than the work function, no electrons are ejected. - Conversely, if the photon energy exceeds this threshold, electrons are released with some kinetic energy.For example, if a light of wavelength 400.0 nm causes a maximum kinetic energy of 1.10 eV, the work function can be deduced using the equation \( KE_{max} = hf_{400} - \phi \). Once this value is known, it can be used to calculate the maximum kinetic energy for light of any other wavelength interacting with the same metal.

photon energy

**Photon energy** is a fundamental element in understanding the photoelectric effect. It is the energy carried by a single photon and is related to its frequency or wavelength. The equation that describes this relationship is \( E = \frac{hc}{\lambda} \), where:- \( E \) is the photon energy.- \( h \) is Planck's constant, approximately \( 6.626 \times 10^{-34} \text{ J·s} \).- \( c \) is the speed of light, around \( 3.00 \times 10^8 \text{ m/s} \).- \( \lambda \) is the wavelength of the light.This means that shorter wavelengths correspond to higher frequency and thus greater photon energy. For instance, light with a wavelength of 300.0 nm has more energy per photon than light with a wavelength of 400.0 nm. In the exercise, calculating photon energy for these specific wavelengths allows one to determine their impact on the maximum kinetic energy of the emitted photoelectrons, while accounting for the constant work function of the metal involved.