Physical Chemistry , 1st ed.

Again, consider that ln xsolvent,solutionln(1 xsolute) xsolute. Making one

final substitution:

xsoluteRTV

This is usually rearranged to read as

VxsoluteRT (7.56)

This equation, which is a remarkable parallel to the ideal gas law, is called the

van’t Hoff equation,after Jacobus van’t Hoff, a Dutch physical chemist who an-

nounced this equation in 1886.* (He was also one of the originators of the con-

cept of the tetrahedral carbon atom, and was the first recipient of the Nobel

Prize for Chemistry in 1901.) The equation relates the osmotic pressure of a

solution to the mole fraction of the solute in the solution. It is strictly valid

only for very dilute solutions (reminiscent of many ideal gas systems), but is

also a useful guide for more concentrated ones.

Example 7.15

What is the osmotic pressure of a 0.010-molal solution of sucrose in water?

If this solution were placed in a system as illustrated in Figure 7.28, how

high would the column of diluted sucrose be at equilibrium if the tube has

a surface area of 100.0 cm^2? Assume 25°C, and that the density of the so-

lution is 1.01 g/mL. Some necessary conversions are 1 bar  105 pascal, and

1 pascal 1 N/m^2 (newton of force per square meter of area), and re-

member that Fmafor converting a mass into its equivalent force. (In this

case,awill be the acceleration due to gravity, which is 9.81 m/s^2 .)

Solution

A 0.010-molal solution contains 0.010 mole of sucrose in 1.00 kg, or 1000 g,

of water. In 1.00 kg of H 2 O, there are 1000 g/(18.01 g/mol) 55.5 mol H 2 O.

Therefore, the mole fraction of sucrose is



55.5

0



.01

0

0

.010

0.000180 xsolute

The molar volume of water is 18.01 mL, or 0.01801 L. Using the van’t Hoff

equation:

(0.01801 L)0.0001800.08314 

m

L

o

b

l

a

K

r

298 K

0.248 bar

This is a substantial osmotic pressure for such a dilute solution! In order to

know how high the column will be, we convert this into N/m^2 :

0.248 bar 

105

1

p

b

a

a

s

r

cals

 

1

1

p

N

a

/

s

m

ca

2

l

2.48 104 N/m^2

For a surface area of 100.0 cm^2 1.00 10 ^2 m^2 , this pressure is caused by

a force determined as

2.48 104 

m

N

 2 1.00 10

 (^2) m (^2) 248 N
198 CHAPTER 7 Equilibria in Multiple-Component Systems
*This is different from the van’t Hoff equation introduced in Chapter 5.