how much energy does a photon carry and how much energy is required to free the most promiscuous electrons.

The energy a photon carries is given by the equation:

[ E = h \nu ]

where:

• ( E ) is the energy of the photon
• ( h ) is Planck's constant ((6.626 \times 10^{-34}) Joule >seconds)
• ( \nu ) (nu) is the frequency of the photon

To determine how much energy is required to free the most readily liberated electron from an atom, we refer to the "work function" of the material. The work function is the minimum energy needed to remove an electron from the surface of a solid (typically a metal). This energy varies depending on the specific material.

For example, the work function for cesium, which has one of the lowest work functions of any metal, is about ( 2.14 ) electron volts (eV). Other materials can have higher work functions; for instance, platinum has a work function of approximately ( 5.65 ) eV.

To illustrate, if we're dealing with a photon intended to eject an electron from cesium, you would need a photon with an energy of at least ( 2.14 ) eV. The frequency (\nu) of the photon required could be calculated from the photon energy equation above, rearranging it:

[ \nu = \frac{E}{h} ]

Substituting the values:

[ \nu = \frac{2.14 , \text{eV}}{4.135 \times 10^{-15} , \text{eV} \cdot \text{s}} ]

[ \nu \approx 5.17 \times 10^{14} , \text{Hz} ]

This frequency corresponds to light in the visible spectrum, specifically near the violet end.