Question

Sodium metal requires a photon with a minimum energy of \(4.41 \times 10^{-19} \mathrm{~J}\) to emit electrons. (a) What is the minimum frequency of light necessary to emit electrons from sodium via the photoelectric effect? (b) What is the wavelength of this light? (c) If sodium is irradiated with light of \(439 \mathrm{~nm}\), what is the maximum possible kinetic energy of the emitted electrons? (d) What is the maximum number of electrons that can be freed by a burst of light whose total energy is \(1.00 \mu \mathrm{J} ?\)

Short answer

(a) The minimum frequency of light necessary to emit electrons from sodium is \(f = 6.66 \times 10^{14} \mathrm{~Hz}\). (b) The wavelength of this light is \(\lambda = 4.51 \times 10^{-7} \mathrm{~m}\) or \(451 \mathrm{~nm}\). (c) The maximum kinetic energy of the emitted electrons is \(K = 1.44 \times 10^{-19} \mathrm{~J}\). (d) The maximum number of electrons that can be freed by a burst of light whose total energy is \(1.00 \mu \mathrm{J}\) is \(N = 2.27 \times 10^6\) electrons.

Step-by-step solution

(a) Find the Minimum Frequency of Light

To find the minimum frequency of light necessary to emit electrons from sodium, we can use the equation of energy in terms of frequency: \[E = h * f\] where E is the energy in Joules, h is the Planck's constant ( \(6.63 × 10^{-34} \mathrm{~J\cdot s}\) ), and f is the frequency in Hertz. We are given the minimum energy required to emit electrons ( \(4.41 × 10^{-19} \mathrm{~J}\) ). Solving for the frequency: \[f = \frac{E}{h}\] \[f = \frac{4.41 × 10^{-19} \mathrm{~J}}{6.63 × 10^{-34} \mathrm{~J\cdot s}}\] Calculate the value of f.

(b) Find the Wavelength of Light

Now that we have the minimum frequency of light necessary to emit electrons, we can find the wavelength of this light using the equation that relates the speed, frequency, and wavelength of light: \[c = f * \lambda\] where c is the speed of light ( \(3.00 × 10^{8} \mathrm{~m/s}\) ), f is the frequency found in part (a), and λ is the wavelength in meters. Solving for the wavelength: \[\lambda = \frac{c}{f}\] Calculate the value of λ.

(c) Find the Maximum Kinetic Energy of Emitted Electrons

We are given that sodium is irradiated with light of 439 nm. First, convert this wavelength into meters: \[439 \, \mathrm{nm} = 439 × 10^{-9}\mathrm{m}\] Next, find the energy of the light by using the equation relating energy, Planck's constant, and the speed of light: \[E = \frac{hc}{\lambda}\] Now, find the maximum kinetic energy (K) of emitted electrons by subtracting the minimum energy required to emit electrons (threshold energy) from the energy of the light: \[K = E - E_{threshold}\] Calculate the value of K.

(d) Maximum Number of Electrons Freed by the Burst of Light

We are given that a burst of light has a total energy of 1.00 µJ ( \(1.00 × 10^{-6}\mathrm{~J}\) ). To find the maximum number of electrons that can be freed by this burst of light, we need to divide the total energy of the burst of light by the minimum energy required to emit one electron (threshold energy): \[N = \frac{E_{total}}{E_{threshold}}\] Calculate the value of N, and round down to the nearest whole number, since we cannot have a fraction of an electron.

Key concepts

Sodium Metal

Sodium metal holds a pivotal role in the study of the photoelectric effect. This phenomenon occurs when light strikes a material, causing it to emit electrons. For sodium, this requires light with a specific minimum energy. Each substance has a different threshold energy, a unique feature based on the material's composition. For example, sodium needs a photon with a minimum energy of \(4.41 \times 10^{-19} \mathrm{~J}\). This threshold not only characterizes the material but also defines the minimum frequency and wavelength necessary for electron emission. As a reactive metal, sodium provides an insightful look into the electromagnetic interactions essential for the photoelectric effect.

Photon Energy

Photon energy is a crucial player in understanding the photoelectric effect. It is defined as the energy carried by a single photon, which is a quantum of light. This energy can be calculated using the formula \(E = hf\), where \(E\) is energy, \(h\) is Planck's constant, and \(f\) is the frequency of the photon.

• This relationship shows that higher frequency photons possess more energy.
• As frequency increases, energy does too, leading to more potential for electron emission from the material.

Practically, for sodium metal, any photon with energy equal to or exceeding the threshold value \(4.41 \times 10^{-19} \, \mathrm{J}\) can cause electron liberation. Understanding photon energy is essential to unraveling how light interacts with matter.

Kinetic Energy of Electrons

The kinetic energy of electrons refers to the energy those electrons possess as they are ejected from the surface of a metal in the photoelectric effect. When discussing the photoelectric effect with sodium, this energy results from the difference between the energy of the incident photon and the threshold energy. The equation used to calculate this is: \[ K = E - E_{threshold} \] where \(K\) represents the kinetic energy, \(E\) is the photon energy, and \(E_{threshold}\) is the minimum energy required to emit electrons.

• This calculation helps determine how much kinetic energy the freed electrons will have.
• The greater the excess energy beyond the threshold, the faster the electrons will move.

This kinetic energy can be observed as current in a photodetector circuit, demonstrating the practical applications of these energy calculations.

Planck's Constant

Planck's Constant is a fundamental physical constant that plays a significant role in quantum mechanics. It has the value \(6.63 \times 10^{-34} \mathrm{~J\cdot s}\). This constant links the energy of photons to their frequency, as demonstrated in the equation \(E = hf\). Planck’s constant is instrumental in determining:

• The frequency required for a photon to exceed the threshold energy and produce the photoelectric effect.
• The energy distribution and behavior of electrons in quantum phenomena.

By employing Planck's constant, scientists gain insights into how energy quantization affects microscopic systems. This constant is foundational in both the study of photons and electron behaviors, making it indispensable to our understanding of the photoelectric effect and broader areas of quantum physics.