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The work function is the minimum energy needed to free an electron from a metal surface, and it corresponds to a threshold frequency phi/h below which no photoemission occurs. A 540 nm threshold implies 2.30 eV, matching potassium. Caesium's low 2.1 eV is why it is used in photocathodes sensitive to visible light.
Work function from threshold
phi = h c / lambda(threshold)
Threshold frequency
f0 = phi / h
Because it depends on how tightly the lattice binds its conduction electrons. Alkali metals hold them loosely; platinum and gold hold them tightly.
Considerably. Oxide layers and adsorbed gases can shift the work function by an electronvolt, which is why photocathodes are made in vacuum.