Calculate the molarity of pure water at \(4.0^{\circ} \mathrm{C}\). The density of water at \(4.0^{\circ} \mathrm{C}\) is \(1.0000 \mathrm{~g} / \mathrm{mL}\).

Understanding the concentration of an aqueous solution is fundamental in chemistry, particularly with regards to solutions where water is the solvent. Concentration refers to the amount of substance known as the solute, that is dissolved in a specific volume of solvent (usually water). The most common unit of concentration in chemistry is molarity, symbolized as M. Molarity is defined as the number of moles of solute per liter of solution. To calculate molarity, you take the total number of moles of solute and divide it by the volume of solution in liters.
For instance, our exercise involves calculating the molarity of pure water, which is a bit unique since the solute and solvent are the same substance. This might seem confusing at first, but it simply means we're considering water in its pure state and focusing on its own molecular makeup, instead of dissolving another substance into it.

Molar mass is an attribute that is specific to every substance, which is the mass of one mole of that substance. The units are typically expressed in grams per mole \(\frac{g}{mol}\). It's a bridge between the macroscopic world (grams) and the microscopic world of molecules (moles).
In the context of our textbook problem, the molar mass of water (H2O) is 18.015\(\frac{g}{mol}\). To calculate the molar mass of a compound like water, you would sum the atomic masses of hydrogen and oxygen, based on the quantity of each in a single molecule of the compound. Since water is composed of two hydrogen atoms and one oxygen atom, its molar mass is the combined atomic weight of these atoms.

Calculating the number of moles of a substance is a key part of solving many problems in chemistry. The formula used to calculate moles is simply the mass of the substance (in grams) divided by its molar mass (in \(\frac{g}{mol}\)).
For example, if we have 1000 g (1 kg) of water, as outlined in our exercise, and we want to know how many moles that corresponds to, we use the molar mass of water, 18.015 \(\frac{g}{mol}\), and apply the formula:
\[Number\,of\,moles = \frac{Mass}{Molar\,Mass}\]
With these calculations, we understand that the molar mass acts as a conversion factor between mass and moles, allowing us to use the observable mass of water to find out how many moles of water molecules we have.