How would you find it?

Question

When a metal was exposed to photons at a frequency of 1.38×1015 s−1, electrons were emitted with a maximum kinetic energy of 3.80×10−19 J.

Calculate the work function, Φ, of this metal.

Φ =

What is the maximum number of electrons that could be ejected from this metal by a burst of photons (at some other frequency) with a total energy of 8.56×10−7 J?

number of electrons:

Answer 1

Φ = 5.34 × 10⁻¹⁹ J  and  1.60 × 10¹² electrons.

Part 1 — the work function

The photoelectric equation is an energy budget. A photon arrives carrying hν. Some of it (Φ) is spent tearing the electron off the metal; whatever is left becomes kinetic energy:

hν = Φ + KE_max  →  Φ = hν − KE_max

hν = (6.626 × 10⁻³⁴ J·s)(1.38 × 10¹⁵ s⁻¹) = 9.14 × 10⁻¹⁹ J

Φ = 9.14 × 10⁻¹⁹ − 3.80 × 10⁻¹⁹ = 5.34 × 10⁻¹⁹ J

(That is 3.34 eV, or 322 kJ/mol — a perfectly ordinary value for a metal, which is a useful reasonableness check.)

Part 2 — and this is the part that is actually being tested

Read the word maximum carefully. You are being asked to spend a fixed budget of 8.56 × 10⁻⁷ J on as many ejections as possible. Every joule that ends up as kinetic energy is a joule wasted — it ejected an electron faster, not an extra electron.

So the most efficient photon is one sitting exactly at the threshold: energy = Φ, KE_max = 0. Then no energy is spent on anything but liberation, and:

n = E_total / Φ = (8.56 × 10⁻⁷ J) / (5.34 × 10⁻¹⁹ J) = 1.60 × 10¹² electrons

This is why the problem says "at some other frequency" — it is quietly telling you that you get to choose the frequency, and the optimal choice is the threshold frequency ν₀ = Φ/h = 8.06 × 10¹⁴ s⁻¹.

Two traps

1. Dividing by hν instead of Φ. Using the original 9.14 × 10⁻¹⁹ J photons gives 9.37 × 10¹¹ — a real answer to a different question ("how many of those photons fit in the budget"), and about 40% low.

2. Thinking two weak photons can gang up on one electron. They cannot. Photoelectric emission is strictly one photon per electron: a photon below ν₀ ejects nothing no matter how many of them arrive. That one-to-one rule is what makes n = E_total/Φ the ceiling rather than just an estimate, and it is the whole reason the photoelectric effect forced physics toward quantised light in the first place.