Rearranging the formula \(C1 \times V1 = C2 \times V2\) to solve for \(V1\):
\[V1 = {{V2 \times C2} \over C1}\]
\[V1 = {{0.04 {μg \over ml} \times 20 \ ml} \over 200 {μg \over ml}}\]
\[V1 = 0.004 \ ml\]
Converting \(0.004 \ ml\) to μl \(= 0.004 \ ml \times {1000 \ μl \over ml} = 4.0 μl\)
So you need to take \(4.0 \ μl\) of the original \({200 \ μg} \over ml\) antibody solution and add it to \(19,996 μl (19.996 ml)\) of diluent. The final \(20 \ ml\) solution will represent a solution of \({0.04 \ μg} \over ml\) of antibody.
Now that we have diluted the antibody we can calculate what volume-to-volume dilution we actually
performed (the dilution factor) because of the relationship \({C1 \over C2} = {V2 \over V1}\):
\[{V1 \over V2} = dilution \ factor\]
\[{20,000 \over 2} = {5,000 \over 1} = 5,000 \ dilution \ factor \ \text{or} \ 1:5,000 \ dilution\]
The dilution factor can also be calculated by dividing the concentration of the starting stock solution by the concentration of the new solution:
\[{C1 \over C2} = \ dilution \ factor\]
\[{{200 \ μg/ml} \over {0.04 \ μg/ml}} = 5,000 \ dilution \ factor \ \text{or} \ 1:5,000 \ dilution\]