The evaporation rate from a clean surface is related to the equilibrium vapour pressure Pe of the evaporating species of molecular weight M by an expression :
\(R=4.43\times10^{-4}\left(\dfrac{M}{T}\right)^{1/2}P_{e}\)
This is the Langmuir-Knudsen relation, and the square root is its signature — option 3:
\(R=4.43\times10^{-4}\left(\dfrac{M}{T}\right)^{1/2}P_{e}\)
Where the square root comes from. The rate at which molecules leave a surface is the rate at which they would strike it from a gas at the equilibrium pressure — the flux given by kinetic theory:
\(\Phi=\dfrac{P_{e}}{\sqrt{2\pi mkT}}\)
The \(\sqrt{mT}\) in the denominator arises because the mean molecular speed varies as \(\sqrt{T/m}\). Converting number flux to mass flux multiplies by the molecular mass, and one power of \(\sqrt{m}\) survives in the numerator — giving the \(\left(M/T\right)^{1/2}\) that identifies the correct option.
| Option | Fault |
|---|---|
| 1 | M/T to the first power — no square root |
| 2 | Right form, wrong constant |
| 3 | ✓ |
| 4 | Both the power and the constant wrong |
Reading the physics out of the formula. Two dependences matter, and they pull in opposite directions.
The dominant term is \(P_{e}\), and it is exponential in temperature through the Clausius-Clapeyron relation, \(P_{e}\propto e^{-\Delta H/RT}\). Raising the source temperature therefore increases the rate enormously — which is how deposition rate is actually controlled.
The explicit \(1/\sqrt{T}\) works the other way but is a very weak dependence, entirely swamped by the exponential. Its presence reflects that hotter molecules, though more numerous in the vapour, are also individually lighter in their contribution per unit pressure.
The heavier the species, the slower it evaporates at a given vapour pressure, since a large M means slow molecules — the \(\sqrt{M}\) in the numerator being a mass-flux effect rather than a count.
Why the relation matters in fabrication. Vacuum evaporation deposits metal films for contacts and interconnect, and the relation gives the deposition rate from the source temperature. It also explains why a high vacuum is essential: the mean free path must exceed the source-to-substrate distance, or the vapour scatters on residual gas and arrives non-uniformly, which is why evaporation is done below \(10^{-5}\) torr.
Hence, the correct expression is option 3.
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