In this example, we calculate the threshold frequency and binding energy from the kinetic energy of an ejected electron
and the wavelength of electromagnetic radiation striking a metal.
Our problem reads, "When a photon with a wavelength of 242 nm
strikes the surface of a metal, the ejected electrons have a kinetic energy of 3.31 electron volts.
Note: 1 electron volt is 1.602 x 10^-19 joules.
Part A: calculate the binding energy of the metal in joules.
Part B: calculate the binding energy in kilojoules per mole.
Part C: calculate the threshold frequency of the metal.
First let's look at the photoelectric effect and the equations and govern it.
In the photoelectric effect, an electron in a metal is absorbing a photon
and is subsequently ejected from the metal if the photon has sufficient energy.
We can calculate the energy of a photon striking the metal with either the frequency or wavelength of the photon.
The energy of the photon equals H C over lambda, which is Planck's constant times the speed of light divided by wavelength.
Or if we have frequency,
the energy of the photon is equal to H nu, which is Planck's constant times frequency.
The electron has some energy keeping it in the metal, which is called the binding energy or phi.
This is the energy that must be overcome to eject the electron from the metal.
If we start with a small energy for the photon striking the metal and slowly increase it,
at some point the photon will have just enough energy to eject the electron.
At this point we're at the threshold frequency: when we just start to eject electrons from the metal.
The energy of the photon striking the metal is Planck's constant times the threshold frequency
which is our equation for binding energy.
If we continue to increase the energy of the photon beyond the threshold frequency, we're adding extra energy to the electron.
Since energy cannot be created or destroyed,
any extra energy beyond the binding energy goes into the kinetic energy of the ejected electron.
So to calculate the kinetic energy, we subtract the binding energy from the energy of the photon.
In part A of our problem, we're asked to find the binding energy of the metal and we're given the kinetic energy of the ejected electron.
We'll need to use the kinetic energy equation.
We're also given the wavelength of the photon, so we'll use H C over lambda to calculate the energy of the photon in that equation.
In part B, we're asked to change the units of the binding energy from joules to kilojoules per mole of electrons.
We can do this using dimensional analysis.
In part C, we're asked to find the threshold frequency of the metal.
We can use our binding energy equation to solve for threshold frequency.
Now that we have a plan let's get started with the math.
For part A, we start with the kinetic energy equation and we solve for binding energy.
Binding energy equals the energy of the photon that strikes the metal subtract the kinetic energy of the electron.
To solve for the energy of photon, we use H C over lamdba because we're given wavelength of the photon.
We can plug in Planck's constant and the speed of light.
When we add the wavelength to this the equation, it needs to be in meters instead of nanometers.
"Nano" means 10 to the negative ninth so 242 nm is 242 times 10 to the negative ninth meters.
Solving this equation, we get 8.209 times 10 to the -19th joules.
We can substitute this into our equation at the top; the next step is to find kinetic energy.
We're given kinetic energy in the problem, but we can't substitute it into the equation in electron volts: we need joules.
To convert from electron volts to joules we use dimensional analysis.
We're given that 1 electron volts equals 1.602 times 10 to the -19th joules in the problem.
Completing the calculation, we get 5.303 times 10 to the -19th joules for the kinetic energy of the electron.
We can substitute this number into our equation at the top.
Solving for binding energy, we get 2.906 times 10 to the -19th joules; this value is the binding energy for a single electron.
In part B, we want to convert the units for the binding energy to kilojoules per mole of electrons.
We can do this using dimensional analysis.
We'll start our dimensional analysis with 2.906 times 10 to the -19th joules per one electron.
We can convert from joules to kilojoules using metric prefixes and from electrons to moles of electrons using Avogadro's number.
Completing the calculation, the binding energy is 175 kilojoules per mole.
Binding energies of metals are often given in units of kilojoules per mole because they're easier to work with.
In part C, we're calculating the threshold frequency from the binding energy.
Rearranging our equation to solve for threshold frequency, we get binding energy divided by Planck's constant.
For this equation we need to use the binding energy in joules for one electron, not kilojoules per mole.
Substituting in the binding energy and Planck's constant, we get 4.39 times 10 to the 14th.
The units are 1 divided by seconds, which are also hertz.