Concise Physical Chemistry

c12 JWBS043-Rogers September 13, 2010 11:27 Printer Name: Yet to Come

PROBLEMS, EXAMPLES, AND EXERCISE 197

by definition and the molar mass of water is 18.02, so we have 997.3/18.02=55.345

mol of water. The mole fractions are

X 2 =

n 2

n 1 +n 2

=

0. 1000

55. 345 + 0. 1000

= 1. 804 × 10 −^3

and

X 1 =

n 1

n 1 +n 2

=

55. 345

55. 345 + 0. 1000

= 0. 9982

Comment:First, notice that the sum of the two mole fractions is exactly 1.000, as

it should be because there are only two components in this solution. Second, notice

that we have already made some pretty serious approximations and assumptions. We

don’t really have the right to assume that the volume taken up by NaCl in solution

is the same as the volume in the crystalline state. Also we are assuming that the

density of water is exactly 1.000 kg dm−^3 and the volume of the solution is exactly

1.000 dm^3 without specifying the temperature. We shall have to take these approxi-

mations more seriously later.

In precise solution chemistry, we often usemolalconcentrations, the amount of

solute expressed in molesper 1000 g of solventin contrast to the ordinary stockroom

unit of moles per liter or molarity. A 0.1000 molal solution contains 1000/18.02=

55.494 moles of water, but so does a 0.2000 molal solution because the volume of

the solute is not subtracted from that of the solvent. That is one advantage of the

molal convention. Another is that the number of grams in a molal solution does

not depend on the temperature, but the number of grams in a molar solution does

because the volume of the solution is temperature-dependent. The mole fraction of the

0.1000molalsolution is

X 2 =

n 2

n 1 +n 2

=

0. 1000

55. 494 + 0. 1000

= 1. 800 × 10 −^3

and

X 1 =

n 1

n 1 +n 2

=

55. 494

55. 494 + 0. 1000

= 0. 9982

with a sum of 1.000 as before.

Why specify a new concentration unit when it comes out the same as the old

one? As the following exercise shows, it does not come out exactly the same and

sometimes it is not even close. For very dilute solutions, the difference is negligible,

which is why the concentrations of solutions in working labs and elementary courses

are usually given in molarity rather than molality. In medicinal and clinical chemistry,

many other concentration units are used but they can all be reduced to the basic ones

with a little algebra.