Question

The work function \((\phi)\) of some metals is listed below. The number of metals which will show photoelectric effect when light of \(300 \mathrm{~nm}\) wavelength falls on the metal is$$ \begin{array}{|l|c|c|c|c|c|c|c|c|c|} \hline \text { Metal } & \mathrm{Li} & \mathrm{Na} & \mathrm{K} & \mathrm{Mg} & \mathrm{Cu} & \mathrm{Ag} & \mathrm{Fe} & \mathrm{Pt} & \mathrm{W} \\ \hline \phi(\mathrm{eV}) & 2.4 & 2.3 & 2.2 & 3.7 & 4.8 & 4.3 & 4.7 & 6.3 & 4.75 \\ \hline \end{array} $$

Short answer

The number of metals showing the photoelectric effect when light of 300 nm falls on them.

Step-by-step solution

Convert Wavelength of Light to Energy

The energy of a photon can be calculated using the equation \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant (\(6.626 \times 10^{-34} \) J s), \( c \) is the speed of light (\(3.00 \times 10^8 \) m/s), and \( \lambda \) is the wavelength of light. Convert the given wavelength (\( \lambda = 300 \) nm) to meters by multiplying by \( 10^{-9} \), and then use the equation to find the energy of the light in joules.

Convert Energy to Electronvolts (eV)

After obtaining the energy in joules, convert it to electronvolts (eV) using the conversion factor \( 1 \) eV = \( 1.602 \times 10^{-19} \) J. This is necessary because the work functions of the metals are given in eV.

Compare Photon Energy with Work Function

Compare the energy obtained in eV for the photon to the work functions of the metals listed in the table. The photoelectric effect will occur for a metal if the photon's energy is greater than the metal's work function. Count the number of metals that meet this criterion.

Key concepts

Work Function of Metals

The work function of a metal is the minimum energy required to release an electron from the surface of the metal. It is a fundamental property that is unique to each type of metal, reflecting the strength with which the electrons are bound to the metal. The work function is often measured in electronvolts (eV), which is a unit of energy commonly used in atomic-scale processes.

When a light wave or photon strikes a metal surface, it can impart energy to an electron. If the energy of the photon is greater than or equal to the work function of that metal, the electron can be ejected in a process known as the photoelectric effect. The list of work functions provided for metals like Lithium (Li), Sodium (Na), and others, serve as a threshold that determines whether a given light source can cause photoelectric emission.

Understanding the work function is crucial for applications like photovoltaic cells, where metals or materials with appropriate work functions are selected to ensure efficiency in converting light into electrical current.

Conversion of Wavelength to Energy

The energy of a photon can be calculated from its wavelength using the formula \( E = \frac{hc}{\lambda} \), where \( h \) is Planck’s constant, \( c \) is the speed of light in a vacuum, and \( \lambda \) is the wavelength. This relationship comes from the theory of quantum mechanics, which describes how light can behave both as a wave and as a particle. Hence, by knowing the wavelength of light, we can determine the energy of the photons it comprises.

As an example, converting a wavelength of 300 nm to energy involves first translating nanometers to meters to align with SI units, and then applying the equation with \( h = 6.626 \times 10^{-34} \) J s and \( c = 3.00 \times 10^8 \) m/s. The result is the energy in joules, which can be converted into electronvolts to compare with work functions, as energy in eV is a more convenient and conventional unit in atomic-scale phenomena.

Energy Threshold in Photoelectric Effect

The energy threshold in the photoelectric effect refers to the minimum energy that incoming photons must possess to eject electrons from a metal surface. This is directly associated with the work function of the metal. If the photon’s energy is below the work function of the metal, the electron cannot be dislodged, and the photoelectric effect will not take place, regardless of the intensity of light.

To observe the photoelectric effect, the energy of the incident light (photon energy) must be greater than the work function (energy threshold) of the metal. It’s a 'quantum leap', meaning that there is no gradual buildup; the electron either absorbs enough energy to overcome the work function or it does not. This concept is pivotal in understanding not only photoelectric systems but also various other phenomenons like threshold frequency and the kinetic energy of ejected electrons, which are central to modern physics and technology.