= Hemocytometer
{wiki}
A <hemocytometer> has a calibrated grid and depth. ATCC describes $1\,\mathrm{mm}^2$ squares $0.1\,\mathrm{mm}$ deep, each sampling $10^{-4}\,\mathrm{mL}$.<Animal Cell Culture Guide by ATCC>[Cell counting, chamber geometry; Figure 2]{ref}
An original constructed <hemocytometer> count uses four such squares after mixing the <biological cell> preparation with the same <volume (thermodynamics)> of stain. The dye-excluding counts are $36,44,40,40$; the dye-stained counts are $4,6,5,5$. The dilution factor is $D=2$. With $v$ the sampled <volume (thermodynamics)> per square and $\bar n$ the <arithmetic mean> count per square,
$$
v=(1\,\mathrm{mm}^2)(0.1\,\mathrm{mm})=10^{-4}\,\mathrm{mL},\qquad c=D\frac{\bar n}{v}.
$$
Hence the undiluted preparation has dye-excluding <number density> $2(40)/10^{-4}=800\,000\,\mathrm{mL}^{-1}$ and total <number density> $2(45)/10^{-4}=900\,000\,\mathrm{mL}^{-1}$. Its dye-excluding fraction is $160/180=8/9\approx88.9\%$. If the well-mixed preparation contains $3\,\mathrm{mL}$, it contains an estimated $2\,400\,000$ dye-excluding <biological cells>. The chamber sample itself contains only $160$ counted dye-excluding <biological cells>.
\Image[/hemocytometer-count-volume-and-dilution.png]
{title=A sampled count becomes a whole-preparation estimate through <volume (thermodynamics)> and dilution}
{description=Original constructed <hemocytometer> example: four $1\,\mathrm{mm}^2$ counting squares at $0.1\,\mathrm{mm}$ depth. Numbers report hypothetical counts, not recorded <HeLa> measurements. Correcting for the equal-volume stain mixture doubles both <number densities> and leaves the dye-excluding fraction unchanged.}
The relation $c=D\bar n/v$ follows from counting material in a known <volume (thermodynamics)>, then undoing dilution. It assumes a representative mixed sample and correctly identified single <biological cells>. Clumps, inconsistent decisions at grid edges, wrong chamber depth and nonrepresentative sampling introduce <measurement error>. A fixed convention such as including top and left edges while excluding bottom and right edges assigns shared-edge centers to just one square. Adding ten times as much suspension to a correctly filled chamber does not make each square's calibrated <volume (thermodynamics)> ten times larger.
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