Question

To resolve details of an object, you must use a wavelength that is about the same size, or smaller, than the details you want to observe. Suppose you want to study a molecule that is about \(1.000 \times 10^{-10} \mathrm{m}\) in length. (a) What minimum photon energy is required to study this molecule? (b) What is the minimum kinetic energy of electrons that could study this? (c) Through what potential difference should the electrons be accelerated to reach this energy?

Short answer

Answer: (a) The minimum photon energy required to study the molecule is \(1.987 \times 10^{-25}\mathrm{J}\). (b) The minimum kinetic energy of electrons required to study the molecule is \(2.426 \times 10^{-18}\mathrm{J}\). (c) The potential difference required to accelerate the electrons to the minimum kinetic energy is \(15.16\mathrm{V}\).

Step-by-step solution

Find the minimum wavelength for studying the molecule

To study an object of length \(1.000 \times 10^{-10} \mathrm{m}\), we need a wavelength smaller or equal to the length of the object. So, the minimum wavelength (\(\lambda_{min}\)) required is: \(\lambda_{min} = 1.000 \times 10^{-10}\mathrm{m}\)

Calculate the minimum photon energy

To find the minimum photon energy, we can use the formula relating energy (\(E\)) and wavelength (\(\lambda\)): \(E = \dfrac{h c}{\lambda}\) where \(h\) is the Planck's constant (\(6.626 \times 10^{-34}\mathrm{Js}\)), \(c\) is the speed of light (\(3.0 \times 10^8\mathrm{m/s}\)), and \(\lambda\) is the minimum wavelength calculated in step 1. Substitute the given values and calculate the minimum photon energy: \(E = \dfrac{(6.626 \times 10^{-34}\mathrm{Js})(3.0 \times 10^8\mathrm{m/s})}{1.000 \times 10^{-10}\mathrm{m}} = 1.987 \times 10^{-25}\mathrm{J}\) So, the minimum photon energy required to study the molecule is \(1.987 \times 10^{-25}\mathrm{J}\).

Calculate the minimum electron kinetic energy

Since we are using electrons to study the molecule, we can use the de Broglie wavelength formula to find the minimum kinetic energy required. The formula for de Broglie wavelength is: \(\lambda = \dfrac{h}{p}\) Here, \(p\) represents the momentum of the electron. To relate momentum to kinetic energy, we can use the formula: \(p = \sqrt{2 m_e K}\) where \(m_e\) is the mass of an electron (\(9.11 \times 10^{-31}\mathrm{kg}\)) and \(K\) is the kinetic energy. Combine these two equations to find the minimum kinetic energy: \(\lambda_{min} = \dfrac{h}{\sqrt{2 m_e K}}\) Solving for \(K\): \(K = \dfrac{h^2}{2m_e\lambda_{min}^2}\) Substitute the given values and calculate the minimum kinetic energy: \(K = \dfrac{(6.626 \times 10^{-34}\mathrm{Js})^2}{2(9.11 \times 10^{-31}\mathrm{kg})(1.000 \times 10^{-10}\mathrm{m})^2} = 2.426 \times 10^{-18}\mathrm{J}\) So, the minimum kinetic energy of electrons required to study the molecule is \(2.426 \times 10^{-18}\mathrm{J}\).

Calculate the potential difference for accelerating the electrons

We can calculate the potential difference (\(V\)) needed to achieve the required kinetic energy using the formula: \(K = eV\) where \(e\) is the elementary charge (\(1.6 \times 10^{-19}\mathrm{C}\)). Solve for \(V\): \(V = \dfrac{K}{e}\) Substitute the calculated minimum kinetic energy and find the potential difference: \(V = \dfrac{2.426 \times 10^{-18}\mathrm{J}}{1.6 \times 10^{-19}\mathrm{C}} = 15.16\mathrm{V}\) So, the potential difference required to accelerate the electrons to the minimum kinetic energy is \(15.16\mathrm{V}\). In summary: (a) The minimum photon energy required to study the molecule is \(1.987 \times 10^{-25}\mathrm{J}\). (b) The minimum kinetic energy of electrons required to study the molecule is \(2.426 \times 10^{-18}\mathrm{J}\). (c) The potential difference required to accelerate the electrons to the minimum kinetic energy is \(15.16\mathrm{V}\).