Why 'max' in $hf=\phi+{1\over{2}}mv_\text{max}^2$? The equation for the photoelectric effect is
$$hf=\phi+{1\over{2}}mv_{\text{max}}^2$$
How does this make sense given that $hf$ describes a single photon and ${1\over{2}}mv_{\text{max}}^2$ describes the maximum kinetic energy over many electrons?
Why can't the photon release many electrons which only just exceed $KE=\phi$? Or, if $hf$ is strictly related to the maximum $KE$, doesn't this mean only one photon is released, in which case the $\text{max}$ subscript doesn't make sense?
 A: Answering your questions one by one:


*

*This equation assumes a monochromatic source of light. So $f$ is same for all photons.

*'Why can't the photons release many electrons which only just exceed hf=ϕ?' this is because not all the energy of photons are transferred to electrons, or some of the electrons may lose kinetic energy via other collisions, etc... Expecting no energy loss is not realistic.

*'Or, if hf is strictly related to the maximum KE, doesn't this mean only one electron is released, in which case the max subscript doesn't make sense?' I don't see how you arrived at your conclusions. Might be you could clarify your logic a bit more, and we can explain it to you.


Cheers.
A: The minimum energy required to eject an electron out of the metal surface is called the work function of the metal. For simplicity let us denote it by $w$ instead of $\phi$.  
When a photon of energy $hv$ (where $h$ is plancks constant and $v$ is frequency) is absorbed by an electron, an amount of energy atleast equal to $w$ (provided $hv$>$w$) is used up in liberating the electron free and the difference $hv-w$ becomes available to the electron as its maximum kinetic energy. Thus,
$$\frac{1}{2}m{v_{max}}^2=hv-w$$
where $m$ is mass of the electron.  
So, yes it is correct that $hv$ is energy of the single photon. And it is also true that kinetic energy given in the above equation refers to the kinetic energy of single electron ejected by absorbing energy of that single photon.  
Photon's energy is quantised, so it's energy can't be fractionally distributed to many electrons to eject more electrons instead of one.
