QUANTUM PHYSICS FUNDAMENTALS - Chapter 2, Exercise 2 Solution
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Finding the Threshold Frequency

PROBLEM
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A metal has a work function of 3.5 x 10^-19 J. Using
KE_max = hf - W, calculate the minimum (threshold) frequency of
light needed to eject any electrons at all (hint: at the threshold,
KE_max = 0).

SOLUTION
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At the threshold frequency, the ejected electron has exactly zero
kinetic energy left over - the photon's entire energy was just
enough to free the electron and no more:

    KE_max = h f - W
    0 = h f - W

Rearranged to solve for the threshold frequency:

    f = W / h

Substitute the given values:

    W = 3.5 x 10^-19 J
    h = 6.626 x 10^-34 J.s

    f = (3.5 x 10^-19) / (6.626 x 10^-34)
    f = 5.28 x 10^14 Hz (approximately)

ANSWER: The threshold frequency is approximately 5.28 x 10^14 Hz.

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WHY THIS WORKS AS AN ANSWER

The threshold frequency is defined precisely as the frequency at
which a photon's energy exactly equals the work function - any lower
frequency photon simply does not carry enough energy to free an
electron at all, no matter how many such photons arrive, which is
exactly the real, observed "hard cutoff" behavior the chapter
describes. Setting KE_max to zero and solving for frequency is the
standard technique for finding this boundary condition, directly
mirroring how a threshold or boundary value is found in many other
physical relationships by setting the varying quantity to its
limiting value.