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What are the units of $N$: the colonies of bacteria or the viable cell count?

This is in regards to the bacterial growth formula used during serial dilutions,

$$ N =N(0) e^{kt} \quad . $$

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Either; they're identical to the units of $N(0)$. In $N =N(0) e^{kt}$,

  • $t$ has units of time (days, minutes, whatever you have chosen)
  • $k$ has units of (1/time) (inverse of $t$'s units)
  • $kt$ is thus unitless, as is $e^{kt}$
  • $N$ and $N(0)$ have the same units.
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If $N$ is a number of something, like anything that is just a number, it is dimensionless and unitless, or with dimension and unit 1.

Note that in chemistry, the "amount" of a substance is not dimensionless but of dimension "amount of matter", with the mole as unit. It is however connected to the number (dimensionless and unitless) of elementary entities in this substance (whether they are atomes, molecules, etc.) through the Avogadro's constant ($N_A = 6,022 140 76 × 10^{23}$ mol$^{–1}$).

However, if your question is whether $N$ and $N(0)$ count the number of individual cells or the number of colonies, I think there is some confusion in your question as the number of colonies (after plating) is a proxi for quantifying the number of viable cells at the moment of plating. But you cannot apply the formula directly to the "number of colonies" since once plated this number will not increase exponentially but stay constant. So $N$ and $N(0)$ refer to the number of (viable) cells.

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