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Photoelectric effect: work function of sodium

Short answer

Work function φ = hc/λ₀ = (6.626×10⁻³⁴×3×10⁸)/(522×10⁻⁹) = 3.81×10⁻¹⁹ J = 2.38 eV. Photon energy for 250 nm: E = 7.95×10⁻¹⁹ J = 4.97 eV. KE = E − φ = 4.14×10⁻¹⁹ J = 2.59 eV.

In the photoelectric experiment, green light, with a wavelength of 522 nm is the longest wavelength radiation that can cause photoemission of electron from a clean sodium surface. Calculate the work function of sodium. If ultraviolet radiation with a wavelength 250 nm is incident to the sodium surface, what will be the kinetic energy of the photo-emitted electrons?

In the photoelectric effect, a photon of energy hf=hc/λhf = hc/\lambda frees an electron only if hf≥ϕhf \ge \phi (work function). Einstein's equation is KEmax=hf−ϕKE_{max} = hf - \phi. The longest wavelength that can cause emission is the threshold wavelength λ0\lambda_0, for which KE=0KE = 0.

Given: λ0=522\lambda_0 = 522 nm, λ=250\lambda = 250 nm, h=6.626×10−34h = 6.626\times10^{-34} J s, c=3×108c = 3\times10^{8} m/s, 1 eV=1.6×10−191\text{ eV} = 1.6\times10^{-19} J.

Work function of sodium

ϕ=hcλ0=6.626×10−34×3×108522×10−9=3.808×10−19 J=3.808×10−191.6×10−19=2.38 eV\begin{aligned} \phi &= \frac{hc}{\lambda_0} = \frac{6.626\times10^{-34}\times3\times10^{8}}{522\times10^{-9}} \\ &= 3.808\times10^{-19}\ \text{J} = \frac{3.808\times10^{-19}}{1.6\times10^{-19}} = 2.38\ \text{eV} \end{aligned}

Energy of the UV photon

E=hcλ=6.626×10−34×3×108250×10−9=7.951×10−19 J=4.97 eV\begin{aligned} E &= \frac{hc}{\lambda} = \frac{6.626\times10^{-34}\times3\times10^{8}}{250\times10^{-9}} \\ &= 7.951\times10^{-19}\ \text{J} = 4.97\ \text{eV} \end{aligned}

Kinetic energy of photoelectrons

KEmax=E−ϕ=4.97−2.38=2.59 eV=4.14×10−19 J\begin{aligned} KE_{max} &= E - \phi = 4.97 - 2.38 = 2.59\ \text{eV} \\ &= 4.14\times10^{-19}\ \text{J} \end{aligned}

Answer: work function of sodium ϕ=3.81×10−19\phi = 3.81\times10^{-19} J =2.38= 2.38 eV; maximum kinetic energy of photoelectrons with 250 nm UV =4.14×10−19= 4.14\times10^{-19} J =2.59= 2.59 eV.

✓ Answer checked by Claude Opus 5.5 (high effort)

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