The metals sodium, iron, and molybdenum have work functions \(2.5 \mathrm{eV}, 3.9 \mathrm{eV},\) and \(4.2 \mathrm{eV},\) respectively. Which of these metals will emit photoelectrons when illuminated with 400 nm light?

The concept of the work function is pivotal in understanding the photoelectric effect. It is defined as the minimum energy required to remove an electron from the surface of a metal. This energy is characteristic of the material, and different metals have different work functions due to their unique electronic structures. For instance, in the given exercise, sodium has a work function of 2.5 eV, while iron and molybdenum have higher work functions of 3.9 eV and 4.2 eV, respectively.

When light shines on a metal's surface, electrons can be ejected only if the photons carry enough energy to overcome the work function. If the energy is lower, no photoelectrons will be emitted, regardless of the intensity or duration of light exposure. This property is critical for devices like photodetectors and solar cells, where material choice is based on the nature of incoming light and desired properties.

Calculating the energy of a photon is fundamental to analyzing the photoelectric effect. Photons, which are particles of light, possess energy that correlates to their wavelength. The relation between the energy (E), Planck's constant (h), the speed of light (c), and the wavelength (λ) is given by the equation:
\( E = \frac{h \times c}{\lambda} \).

In our given problem, using the wavelength of 400 nm, we utilize this formula to calculate the energy of the photon in Joules. However, since the work function of metals is often expressed in electron-volts (eV), conversion is necessary for comparison. By using the conversion factor where \(1 \mathrm{eV} = \frac{1 \mathrm{J}}{1.602 \times 10^{-19}} \), we matched the units to the work functions. Such calculations are crucial for determining the outcome of photoelectric experiments and applications.

Planck's constant (h) is a fundamental constant that appears in many equations in quantum physics, representing the quantization of energy. Its value is approximately 6.626 x 10^-34 Joule seconds (Js). Planck's constant bridges the gap between the macroscopic classical mechanics and the microscopic quantum world.

This constant is central to the equation for photon energy and the photoelectric effect. When we utilize the constant to calculate photon energy, it allows us to predict whether a particular light wavelength will cause electron emission from a metal surface. The precision of Planck's constant is crucial because a small error in its value can lead to significant discrepancies in experimental results and theoretical predictions.