Serotonin (C\text{C}_{10}\text{H}_{12}\text{N}_{2}\text{O}) ; molar mass = 176 g mol$^{-1}$) is a compound that conducts nerve impulses in the brain and muscles - VCE - SSCE Chemistry - Question 2 - 2008 - Paper 1

To find the number of molecules of serotonin in one millilitre of spinal fluid, we start with the given concentration:

1.5 ng L$^{-1}$ means there are 1.5 nanograms of serotonin in one litre of spinal fluid.

Next, convert nanograms to grams:

1.5 ng = 1.5 imes 10^{-9} g

Using the molar mass of serotonin, which is 176 g mol$^{-1}$, we can find the number of moles in 1.5 ng:

$ext{Number of moles} = \frac{1.5 \times 10^{-9} g}{176 g mol^{-1}} \approx 8.52 \times 10^{-12} mol$

Next, to find the number of molecules, we use Avogadro's number ( ext{N}_{A} = 6.022 \times 10^{23} molecules mol$^{-1}$):

$\text{Number of molecules} = \text{Number of moles} \times \text{N}_{A}$

Substituting the values, we find:

$\text{Number of molecules} = 8.52 \times 10^{-12} mol \times 6.022 \times 10^{23} molecules mol^{-1} \approx 5.13 \times 10^{12} molecules$

Since this value is per litre, and we are looking for the amount in one millilitre, we divide by 1000:

$\text{5.13 x 10}^{12} molecules L^{-1} \div 1000 = 5.13 x 10^{9} molecules mL^{-1}$

Therefore, the correct answer would be option A: 5.13 x 10$^{9}$.