Points to Ponder

The phrase "free electrons" in a metal is an approximation. These electrons can move freely inside the metal because they experience a roughly constant electric potential there. However, they are not free to leave the metal; escaping requires extra energy to overcome the surface barrier.

At any given temperature, the free electrons in a metal do not all possess the same kinetic energy. Instead, they have a specific energy distribution, analogous to how gas molecules have a speed distribution. Crucially, this electron distribution differs from the Maxwell-Boltzmann distribution you studied in kinetic theory because electrons obey the Pauli exclusion principle — no two electrons can occupy the same quantum state.

Because of this energy distribution, different electrons need different amounts of additional energy to escape the metal. An electron that already has high kinetic energy needs less extra energy to leave than one with low kinetic energy. The work function ($\phi_0$) is defined as the minimum energy required for any electron to escape — it is the energy needed by the highest-energy electron at absolute zero.

Photoelectric experiments show that during light–matter interaction, energy is absorbed in discrete packets of size $h\nu$. This is a statement about the absorption process, not necessarily a claim that light itself is made of particles. The photon picture (light as particles) is a stronger interpretation that goes beyond this observation. …