Question

The work function of an element is the energy required to remove an electron from the surface of the solid element. The work function for lithium is \(279.7 \mathrm{kJ} / \mathrm{mol}\) (that is, it takes \(279.7 \mathrm{kJ}\) of energy to remove 1 mole of electrons from 1 mole of Li atoms on the surface of Li metal; 1 mol \(L i=6.022 \times\) \(10^{23}\) atoms Li). What is the maximum wavelength of light that can remove an electron from an atom on the surface of lithium metal?

Short answer

To find the maximum wavelength of light that can remove an electron from an atom on the surface of lithium metal, we first convert the work function to J/atom: \(\frac{279.7 \times 10^3\ \mathrm{J/mol}}{6.022 \times 10^{23}\ \mathrm{atoms/mol}}\). Then, we use Planck's equation and the speed of light equation to determine the wavelength: \(\lambda = \frac{c}{\nu} = \frac{c}{\frac{E\ (\mathrm{photon})}{h}}\), where E (photon) is the work function per atom, c is the speed of light, and h is Planck's constant. After calculating the wavelength, convert it to an appropriate unit such as nanometers if necessary.

Step-by-step solution

Convert the work function from kJ/mol to J/atom

Before we proceed with the calculations, let's first convert the work function from kJ/mol to J/atom, which will be more useful for our calculations. Work function \(= 279.7\ \mathrm{kJ/mol} = 279.7 \times 10^3\ \mathrm{J/mol}\) We need to convert this value to J/atom. Since 1 mole of Li consists of \(6.022 \times 10^{23}\) atoms, we can find the work function per atom by dividing the work function in J/mol by the number of atoms in 1 mole: Work function per atom \(= \frac{279.7 \times 10^3\ \mathrm{J/mol}}{6.022 \times 10^{23}\ \mathrm{atoms/mol}}\)

Calculate the energy of the photon required to remove an electron

Now that we have the work function per atom, we can use the work-energy relationship to determine the energy (E) of the photon required to remove an electron from the Li atom: E (photon) \(= \mathrm{Work\ function\ per\ atom}\)

Calculate the frequency of the photon using Planck's equation

Planck's equation relates the energy of a photon with its frequency (ν): E (photon) \(= h\nu\) where h is the Planck's constant (\(6.626 \times 10^{-34}\ \mathrm{J\cdot s}\)). We can calculate the frequency (ν) of the photon using the equation: ν \(= \frac{E\ (\mathrm{photon})}{h}\)

Calculate the wavelength of the photon using the speed of light

To find the wavelength (λ) of the photon, we can use the equation relating the speed of light (c), frequency (ν) and wavelength (λ): c \(= \nu \lambda\) where c is the speed of light (\(2.998 \times 10^8\ \mathrm{m/s}\)). We can now solve for the wavelength (λ): λ \(= \frac{c}{\nu}\)

Calculate the maximum wavelength of light capable of removing an electron from an atom on the surface of lithium metal

Following the steps above, we can calculate the maximum wavelength of light that can remove an electron from an atom on the surface of lithium metal by plugging in the values from our previous calculations. This will give us the final answer in meters. Note: Make sure to convert wavelength to an appropriate unit, such as nanometers, if necessary.

Key concepts

Quantum Mechanics

Quantum Mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It introduces a significant departure from classical physics, primarily at the quantum realm of very small scales, where the weird and counterintuitive phenomena of wave-particle duality take reign. Electrons in a quantum mechanical system do not inhabit fixed orbits but exist in states described by mathematical wave functions. These functions can only provide probabilities for where an electron might be found.

One crucial aspect of quantum mechanics is that energy is quantized. This means that energy is absorbed or emitted in discrete units called quanta or photons in the context of light. Quantum mechanics paves the way for understanding the interactions of matter and light, which leads us to critical phenomena like the photoelectric effect.

Photoelectric Effect

The photoelectric effect is an essential quantum mechanical phenomenon where electrons are ejected from the surface of a material when it is exposed to light of sufficient energy. This discovery was key in underscoring the particle-like behavior of light and supports the concept that light can be thought of as a stream of particles, or photons. Each photon carries a discrete amount of energy.

The work function, in relation to the photoelectric effect, is the minimum energy required to remove an electron from the surface of a material. If the incoming photon has more energy than the work function of the material, an electron can be emitted. This process is a vivid display of energy quantization in action and is directly tied to photon energy calculation.

Planck's Equation

Planck's equation is a fundamental formula in quantum mechanics expressed as

\( E = h u \)

where \( E \) is the energy of a photon, \( u \) is its frequency, and \( h \) is Planck's constant, approximately \( 6.626 \times 10^{-34} \mathrm{J\cdot s} \). This equation emphasizes that the energy of a photon is directly proportional to its frequency and represents the quantized nature of energy in the realm of the small.

Understanding Planck's equation is instrumental in calculations involving the interaction between light and matter, such as determining the frequency of light needed to overcome the work function of a material in the photoelectric effect.

Photon Energy Calculation

Photon energy calculation is an essential application of Planck's equation in identifying the energy that a photon carries. The process involves calculating the energy based on the frequency of the light. Since we often deal with light in terms of wavelength, there is an inverse relationship between the wavelength of light and its frequency, with longer wavelengths corresponding to lower frequencies and therefore lower energy.

A practical application of photon energy calculation is the determination of the maximum wavelength of light capable of ejecting an electron off a material's surface. By calculating this maximum wavelength, one can understand the material's responsiveness to different light and make predictions about how it interacts with electromagnetic radiation across the spectrum.

The steps provided in the exercise use photon energy calculations to determine the maximum wavelength of light that can eject an electron from lithium metal. This calculation is key to understanding experimental outcomes in both theoretical and applied sciences.