Density is a very important concept in physics. It tells how much mass is packed

Question

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 1023) and that a liter means 1000 cm3 (since one thousandth of a liter -- one milliliter = 1 cm3), it's reasonably straightforward to convert from moles/liter to molecules/cm3. Let's write that m = molarity in units of moles/liter and n = number density = N/V, the number of molecules divided by the volume they are in, in units of molecules/m3. Since m and n must be proportional, we expect there is an equation

n = m

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

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. 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.

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. The distance,

d = s1/3

is a reasonable estimate for the average separation between molecules of molarity m. This doesn't have an official name, but we might call it separateness.

B. Given that the molecular weight of water (H2O) is equal to 18 D, and that the density of water is 1 gram/ml, find the molarity, specific volume, and separateness of water. (Note the units you are supposed to use.)
m = ____
s = ____ nm3
d = ____ nm


The numbers -- specific volume and separateness can give valuable insight to the question, What drives diffusion? In our reading, Diffusion and random walks, we stated that individual molecules in fluids undergo a random walk, moving in unpredictable ways. Yet somehow, there emerges from these random motions a flow from higher to lower concentration. (See Fick's Law.)

One potential way of thinking about diffusive flow is that diffusing molecules collide with each other more often in regions of high concentration and are "driven away" towards regions of low where they don't have as many collisions. If this is true, diffusing molecules would have to be reasonably close to each other. Let's consider this in a realistic case, using our method of estimating number density and the distance between molecules.

D. Compare these to the specific volume and separation distances you found for the molecules of water and discuss whether you think that the jiggling motion of the acetylcholine molecules are primarily driven by collisions with the jiggling water molecules or by collisions with each other?

Explanation / Answer

A)

To change the units on m from moles/ml to molecules/m3, we have to multiply by the correct conversion factors (the proper forms of "1"). Since

1 mole = 6.022 x 1023 molecules

and

1 liter = 1000 cm3 = 103 x (10-2 m)3 = 10-3 m3,

we get,

n = m[(6.022*1023 molecules)/ mole] * (1 liter/ 10-3 m3)

= [(6.022*1026 molecules-liter)/ (mole-m3)] * m

and must conclude that is the value in the parenthesis,

=> = 6.022 x 1026 molecules-liter/mole-m3

B)

Since one liter is 1000 ml, one liter of water will have a mass of 1000 grams (1 kg). It will therefore have a molarity of

m = (1000 g)/(18 g/mole) = 55.6 moles

It's specific volume will be

s = 1/m = 1/(6x1026x55.6) m3/molecule

= 0.30 x 10-27 m3/molecule

= 0.30 nm3/molecule

and its separateness will be

d = s1/3 = 0.67 nm

C)

It's specific volume will be

s = 1/m = 1/(6x1026x10-4) m3/molecule

= 16 x 10-24 m3/molecule

= 16,000 nm3/molecule

And its separateness will be

d = s1/3 = 25 nm

D)

The acetylcholine molecules are on the average 50 times (25/0.5 = 50) further apart in each direction than are the water molecules. So in a volume that holds 1 acetylcholine molecule there are about 503= 125,000 water molecules. Each acetylcholine molecule is going to collide with water molecules almost all of the time and only very rarely with another acetylcholine molecule. Almost all of its jiggling will be driven by collisions with water molecules.

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