Counting by Weighing and Avogadro number

The size of molecule is so small that it is physically difficult if not impossible to directly count out molecules (Figure .1). However. we can count them indirectly by using a common trick of counting by weighing.

Figure .1 : A comparison of the scales of various biological and technological objects.

Consider the example of counting nails in a big box at a hardware store. You need to estimate the number of nails in a box. The weight of an empty box is 213 g and the weight of the box plus a bunch of big nails is 1340 g . Lets assume we know that the weight of one big nail is 0.450 g. I hope you arent going to tear open the package and count the nails. We agree that

mass of big nails=1340g−213g=1227g

Therefore

You have just counted the number of big nails in the box by weighing them (rather than counting them individually).

Figure 2: Galvanized nails. Individually counting nails in a box would would require significant effort. Alternatively we can count them by weighing.

Now consider if the box of nails weighed the same, but box were filled with small nails with an individual mass of 0.23g/small nail instead? You would do the same math, but use a different denominator in below Equation :

The individual mass is the conversion factor used the calculation and changes based on the nature of the nail (big or small). Lets ask ask a different question? How many dozens of nails are their in that same box of small nails described above.

If we knew the information from in below Equation , we can just use the conversion of how many nails is in a dozen

If we wanted to get this value from weighing we use the dozen mass instead of individual mass of

So following Equation  we get