Question

White light, \(\lambda=400,\) to \(750, \mathrm{nm},\) falls on barium \((\phi=2.48 \mathrm{eV})\) a) What is the maximum kinetic energy of electrons ejected from the metal? b) Would the longest-wavelength light eject electrons? c) What wavelength of light would eject electrons with zero kinetic energy?

Short answer

b) Can the longest-wavelength light eject electrons? c) What is the wavelength of light that would eject electrons with zero kinetic energy? a) The maximum kinetic energy of electrons ejected from the metal is 0 eV since no electrons are ejected. b) No, the longest-wavelength light cannot eject electrons. c) The wavelength of light that would eject electrons with zero kinetic energy is 505.4 nm.

Step-by-step solution

Calculate the energy of a photon

To determine the energy of a photon, we can use the equation: \(E = h \cdot \dfrac{c}{\lambda}\). We are given the range in which \(\lambda\) lies as \(400\,\text{nm}\) to \(750\,\text{nm}\). So first, we need to convert the wavelength from nanometers into meters: \(\lambda_\text{min}=400\,\mathrm{nm}= 400 \times 10^{-9}\,\mathrm{m}\) and \(\lambda_\text{max}=750\,\mathrm{nm}= 750 \times 10^{-9}\,\mathrm{m}\).

Calculate the maximum kinetic energy of ejected electrons

Use the photoelectric effect equation: \(K_\text{max} = hv/\lambda - \phi\). Here, \(h = 6.63 \times 10^{-34} \, \mathrm{m^2\cdot kg/s}\) is the Planck's constant, \(c = 3 \times 10^8\, \mathrm{m/s}\) is the speed of light and the work function \(\phi = 2.48\, \mathrm{eV} = 2.48 \times 1.6\times10^{-19} \, \mathrm{J}\). To find the maximum kinetic energy of ejected electrons, we need to find the maximum energy of photons, which corresponds to the minimum wavelength (\(\lambda_\text{min}\)). \(K_\text{max} = h \cdot \dfrac{c}{\lambda_\text{min}} - \phi = \frac{6.63 \times 10^{-34}\, \mathrm{m^2\cdot kg/s}\cdot 3 \times 10^8\, \mathrm{m/s}}{400 \times 10^{-9}\,\mathrm{m}} - 2.48 \times 1.6\times10^{-19} \, \mathrm{J} = 3.12\times 10^{-19}\, \mathrm{J}- 3.97\times 10^{-19}\, \mathrm{J} = -0.85\times 10^{-19}\, \mathrm{J}\). Since the maximum kinetic energy is negative, electrons are not ejected from the metal. a) The maximum kinetic energy of electrons ejected from the metal is 0 eV since no electrons are ejected.

Determine if the longest-wavelength light can eject electrons

As calculated above, the photon energy of the longest-wavelength light does not reach the work function. Therefore, no electrons are ejected from the metal. b) No, the longest-wavelength light cannot eject electrons.

Calculate the threshold wavelength for zero kinetic energy

To find the threshold wavelength for zero kinetic energy, we have to solve the photoelectric effect equation for \(\lambda\) when \(K_\text{max}=0\): \(0 = h\cdot \dfrac{c}{\lambda} - \phi \Rightarrow \lambda = \frac{h\cdot c}{\phi}\). \(\lambda = \dfrac{6.63 \times 10^{-34}\, \mathrm{m^2\cdot kg/s}\cdot 3 \times 10^8\, \mathrm{m/s}}{2.48 \times 1.6\times10^{-19} \, \mathrm{J}} = 505.4 \times 10^{-9}\, \mathrm{m} = 505.4\, \mathrm{nm}\). c) The wavelength of light that would eject electrons with zero kinetic energy is 505.4 nm.

Key concepts

Work Function

The work function is a fundamental concept in the study of the photoelectric effect. It represents the minimum energy required to remove an electron from the surface of a metal. When light shines onto a metal surface, if the energy of the photons is equal to or greater than the metal's work function, electrons can be ejected; this phenomenon is known as photoelectric emission.

The work function, usually denoted as \( \phi \), is measured in electronvolts (eV) or joules (J). For barium, in our textbook exercise, the work function is given as \( \phi = 2.48 \, \text{eV} \). In order to determine whether electrons will be ejected, we compare the photon energy of the incident light to this work function value. If the photon energy is lower than \( \phi \), there won't be any electron ejection, answering part 'b' of the exercise directly.

Kinetic Energy of Electrons

When discussing the photoelectric effect, kinetic energy refers to the energy possessed by ejected electrons as a result of absorbing photons. This energy is given by the equation \( K_\text{max} = hu - \phi \), where \( K_\text{max} \) is the maximum kinetic energy, \( h \) is Planck's constant, \( u \) is the frequency of the incident light, and \( \phi \) is the work function of the metal.

It's important to note that only the excess energy of the photons over the work function contributes to the kinetic energy of the electrons. Hence, when the photon energy is exactly equal to the work function, the ejected electron will have zero kinetic energy, as shown in part 'c' of the exercise. If the energy of the photons is less than the work function, then no electrons are ejected, and thus, the kinetic energy of the electrons is zero, as indicated in part 'a' of the exercise.

Threshold Wavelength

Threshold wavelength is another crucial concept related to the photoelectric effect. It is defined as the maximum wavelength of light that can still eject electrons from the metal. Above this wavelength, the photons' energy is too low to overcome the work function, and no photoelectric emission occurs.

The threshold wavelength, denoted as \( \lambda_\text{threshold} \), is inversely proportional to the work function, which can be expressed through the equation \( \lambda_\text{threshold} = \frac{hc}{\phi} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \phi \) is the work function. Applying this to our exercise, we derived that the threshold wavelength for barium, which has a work function of \( 2.48 \, \text{eV} \), is \( 505.4 \, \text{nm} \). Any wavelength longer than \( 505.4 \, \text{nm} \) does not provide enough energy to eject electrons. This value helps us answer part 'c', as it is the exact boundary between light that can and cannot cause electron ejection.