Physical Chemistry Third Edition

10 1 The Behavior of Gases and Liquids

Solution

The Celsius temperature differs from the absolute temperature by 273.15 K, but the Celsius

degree is the same size as the kelvin.

T 273 .15 K+ 0. 00 ◦C 273 .15 K

We convert amount of gas to moles and the volume to m^3 using the factor-label method:

n(20.00 g)

(

1 mol

20 .179 g

)

 0 .9911 mol

V(22.400 L)

(

1m^3

1000 L

)

 0 .022400 m^3

We now carry out the numerical calculation:

P

nRT

V



( 0 .9911 mol)

(

8 .314 J K−^1 mol−^1

)

( 273 .15 K)

(

0 .022400 m^3

)

 1. 005 × 105 Jm−^3  1. 005 × 105 Nm−^2  1. 005 × 105 Pa

You can see how the symbolic formula is used as a template for setting up the numerical

calculation. The unit conversions can also be included in a single calculation:

P

(

20 .00 g

)(

8 .314 J K−^1 mol−^1

)

( 273 .15 K)

( 22 .400 L)

(

1 mol

20 .179 g

)(

1000 L

1m^3

)

 1. 005 × 105 Jm−^3  1. 005 × 105 Nm^2  1. 005 × 105 Pa

The pressure can be expressed in atmospheres by another conversion:

P(1. 005 × 105 Pa)

(

1 atm

101325 Pa

)

 0 .9919 atm

A calculator displayed 100,486.28725 Pa for the pressure in the previous example.

The answer was then rounded to four digits to display onlysignificant digits. In car-

rying out operations with a calculator, it is advisable to carry insignificant digits in

intermediate steps in order to avoid round-off error and then to round off insignificant

digits in the final answer. You can review significant digits in any elementary chemistry

textbook. The main idea is that if the calculation produces digits that are probably incor-

rect, they are insignificant digits and should be rounded away. An important rule is that

in a set of multiplications and divisions, the result generally has as many significant

digits as the factor or divisor with the fewest significant digits.

Another important technique in problem solving is to figure out roughly how large

your answer should be and what its units should be. For example, the author had a

student under time pressure in an examination come up with an answer of roughly

1030 cm for a molecular dimension. A moment’s thought should have revealed that this

distance is greater than the size of the known universe and cannot be correct. Many

common mistakes produce an answer that either has the wrong units or is obviously

too large or too small, and you can spot these errors if you look for them. You should

always write the units on every factor or divisor when setting up a numerical calculation

so that you will be more likely to spot an error in units.