4.3.3.P10

Density is a very important concept in physics. It tells how much mass is packed into a unit volume. But in chemistry, since atoms interact individually and not according to mass, the "density" that matters is the count -- the number of atoms of a particular chemical per unit volume -- a number density. In this problem we'll explore how a number density gives us information on how close molecules are to each other -- how much space each individual molecule gets. This gives us a good insight to how to think differently about gases and liquids on the molecular level.

A. The 'number density" in chemistry is *molarity: *the number of moles of a particular chemical in a given volume. The particular choice made in chemistry is typically the number of (gram) moles of a particular chemical in a liter. Physicists tend to be more interested in molecules and in metric units. Given that a mole of a chemical means Avogadro's number of molecules (6 x 10^{23}) and that a liter means 1000 cm^{3} (since one thousandth of a liter -- one milliliter = 1 cm^{3}), it's reasonably straightforward to convert from moles/ml to molecules/cm^{3}. Let's write that *m* = molarity in units of moles/ml and *n* = number density *= N/V, *the number of molecules divided by the volume they are in, in units of molecules/m^{3}. Since *m* and *n* must be proportional, we expect there is an equation

*n* = α*m*

where α is some constant. Find the numerical value of α and its units (keeping "moles" and "molecules" as units).

The bigger *n* is, the closer together are the molecules. It's useful to also consider the reciprocal of this -- the volume divided by the number of molecules. This gives the amount of volume each molecule occupies by itself, on the average. ("By itself" meaning with no other molecules of the same type. Of course there may be other molecules in this volume, but on the average it gives a sense of how far apart each molecule is from others of its kind.) This quantity, *s* = 1/*n* = *V/N*, is called *specific volume*, or we might call it *separateness*. Larger *s* means that each molecule occupies more volume so they are more separated, while smaller *s* means each molecule occupies less volume so they are closer (less separated). Note that here "occupies" means "moves through by itself", not the actual volume the molecule takes up.

B. In chemistry we learn that at STP (standard temperature and pressure), 22.4 liters of gas contains one mole of molecules. Given that air has a density of 1.225 kg/m_{3}, and it is about 20% oxygen (O_{2}) and 80% nitrogen (N_{2}), find the molarity and specific volume for molecular oxygen and nitrogen in the air. An oxygen molecule has a molecular weight 0f 32 D and nitrogen has a molecular weight of 28 D.

C. Given that the molecular weight of water (H_{2}O) is equal to 18 D, and that the density of water is 1 gram/ml, find the molarity and specific volume of water.

D. Given that the molecular diameter of a one molecule of molecular oxygen, molecular nitrogen, and water are all about 0.12 nm, what fraction of their individual volumes does each of them fill?

E. If we want a sense of how far apart molecules are, it's more convenient to have a distance, not a volume. To get a distance (a length) from a volume (a length cubed), we need to take a cube root. Is the distance,

*d = s*^{1/3}

where *s = *1*/α**m* a reasonable estimate for the average separation between molecules of molarity *m*? Discuss why you do or do not think so.

F. Find the values of *d* for oxygen and nitrogen molecules in air at STP and for liquid water.

Joe Redish and Marco Colombini

11/19/15

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