Page 43 - Theory and Problems of BEGINNING CHEMISTRY

32 MATHEMATICAL METHODS IN CHEMISTRY [CHAP. 2

2.56. Round each of the following calculated results to the nearest integer (whole number): (a) 1.545 cm, (b)
5.500 cm, (c) 5.5 cm, (d) 4.501 cm, (e) 45.95 cm, and (f) 4.500 cm.
Ans. (a)2cm (b)6cm (c)6cm (d)5cm (e)46cm (f) 5 cm (or 4 cm with the sophisticated system).

2.57. Add, expressing the answer in the proper number of significant figures:
(a) 2000.0cm + 0.02 cm
3
(b)2.0000 × 10 cm + 2 × 10 −2 cm
Ans. (a) 2000.0cm
0.02 cm
2000.02 cm → 2000.0cm
The fraction added is so small relative to the first number that it does not make any difference within
the number of significant figures allowed.
(b) This part is exactly the same as part (a). The same numbers, expressed in scientific notation with
the same number of significant digits, are used. Be aware that when a small number is added to a
much larger one, the small number may not affect the value of the larger within the precision of its
measurement.

2.58. Determine the answers to the following expressions to the correct number of significant digits:
(a) 18.00 mL(2.50 g/mL) (c) 167 cm − 15.3cm
(b)0.207 m − 9.2mm (d) 9.00 g
1.50 mL
Ans. (a) 18.00 mL(2.50 g/mL) = 45.0 g (three significant digits)
(b)0.207 m − 9.2mm = 0.207 m − 0.0092 m = 0.198 m (three decimal places)
(c) 167 cm − 15.3cm = 151.7cm → 152 cm
9.00 g
(d) = 6.00 g/mL (three significant digits)
1.50 mL
2.59. The radius of a circle is 3.00 cm. (a) What is the diameter of the circle? (d = 2r) (b) What is the area of
2
the circle? (A = πr )
Ans. (a) The diameter is twice the radius:

d = 2r = 2(3.00 cm) = 6.00 cm
The 2 here is a definition, not a measurement, and does not limit the accuracy of the answer.
2
2
(b) A = πr = (3.141 59)(3.00 cm) = 28.3cm 2
We can use the value of π to as many significant digits as we wish. It is not a measurement.
2.60. Perform the following calculations to the proper number of significant digits:
8
7
5
(a) (1.20 × 10 g + 9.7 × 10 g)/(3.04 × 10 L)
4
3
(b) (1.66 × 10 −3 m + 6.78 × 10 −2 m)(4.1 × 10 m + 1.3 × 10 m)
9
9
4
(c) (7.7 × 10 m + 3.3 × 10 m) − (9.4 × 10 m)
23
(d) (6.02 × 10 atoms + 6.02 × 10 21 atoms) − (3.95 × 10 22 atoms)
(e)1 cm − (1.6 × 10 −6 cm + 1.1 × 10 −7 cm)
2
Ans. (a)7.14 × 10 g/L (The sum has three significant digits.)
2
3
(b)2.9 × 10 m (The second sum limits the answer to two significant digits.)
(c)1.10 × 10 10 m (The sum has three significant digits, and the number subtracted is too small to affect
the sum at all.)
23
(d)5.69 × 10 atoms
(e) 1 cm (The sum is too small to affect 1 cm at all.)