1.2 Systems and States in Physical Chemistry
19
The volume of the object is equal to L 3 , so
V (T + dT ) L(T )
3
1 + α L dT
3
L(T )
3
1 + 3α L dT + 3(α L dT )
2
+ (α L dT )
3
(1.2-20)
Since dT is small, the last two terms are insignificant compared with the term that is
proportional to dT .
V (T + dT ) L(T )
3 (1 + 3α L dT )
(1.2-21)
The volume at temperature T + dT is given by
V (T + dT ) V (T ) +
∂V
∂T
dT V (T )(1 + αdT )
(1.2-22)
Comparison of Eq. (1.2-22) with Eq. (1.2-21) shows that
α 3α L
(1.2-23)
This relationship holds for objects that are not necessarily shaped like a cube.
E X A M P L E 1.7
The linear coefficient of expansion of borosilicate glass, such as Pyrex or Kimax , is equal
to 3.2 × 10 −6 K −1 . If a volumetric flask contains 2.000000 L at 20.0 ◦ C, find its volume at
25.0 ◦ C.
Solution
V (25 ◦ C) V (20 ◦ C)(1 + 3α L (5.0 ◦ C))
(2.000000 L)(1 + 3(3.2 × 10 −6 )(5.0 ◦ C)) 2.000096 L
Exercise 1.5
Find the volume of the volumetric flask in Example 1.7 at 100.0 ◦ C.
Moderate changes in temperature and pressure produce fairly small changes in the
volume of a liquid, as in the examples just presented. The volumes of most solids
are even more nearly constant. We therefore recommend the following practice: For
ordinary calculations, assume that liquids and solids have fixed volumes. For more
precise calculations, calculate changes in volume proportional to changes in pressure
or temperature as in Examples 1.5 and 1.6.
Exercise 1.6
The compressibility of acetone at 20 ◦ C is 12.39 × 10 −10 Pa −1 , and its density is 0.7899 g cm −3
at 20 ◦ C and 1.000 bar.
a. Find the molar volume of acetone at 20 ◦ C and a pressure of 1.000 bar.
b. Find the molar volume of acetone at 20 ◦ C and a pressure of 100.0 bar.