Compound CalculationsThe Ideal Gas LawIn an earlier physical science course, you may have learned that, for most gases near normal temperatures and pressures, a certain amount of the gas, its pressure, its temperature, and its volume are all related by a constant called the universal gas constant, R, in a formula representing the ideal gas law. The constant is calculated at standard conditions using the formula

,

where P is pressure, V is volume, n is the number of moles (the amount), and T is the absolute temperature.Calculate R if P = 101 325 Pa, V = 0.0224 m3, n = 1.00 mol, and T = 273.15 K.

SolutionFormula:

Given: P = 101 325 PaV = 0.0224 m3

n = 1.00 molT = 273.15 K

This compound calculation consists of all multiplication and divi-sion. Applying Math Rule 3, note that the factors with the fewest SDs have 3 SDs. Therefore, the rounded result may not have more than 3 SDs. Also note that none of the original units in the example prob-lem canceled during this simultaneous calculation. They all appear in the final result—the universal gas constant.

Sum of AreasTwo students are assigned the task of determining the area of the working tabletops in the school’s laboratory. The first student finds the dimensions of the students’ lab tabletop to be 8.4 m long by 0.76 m wide. The other student measures the teacher’s demonstra-tion tabletop and reports that it is 189.2 cm long and 91.4 cm wide. What is the total tabletop working area in the laboratory in square meters?

SolutionGiven: Let ls and ws be the length and width of the students’ tabletop. Let lt and wt be the length and width of the teacher’s demonstration desktop. The given measurements are:

RPVnT

=

RPVnT

R

= =

=

( )( . )( . )( . )

101325 0 02241 00 273 15

101325

Pa m mol K

3

PPa m mol K

3

Pa mmol K

P

3

0 02241 00 273 15

8 309

8 31

.. .

.

.

R

R

= ⋅ ⋅= aa m

mol K3

SDs allowed)⋅ ⋅ (3

RPVnT

R

= =

=

( )( . )( . )( . )

101325 0 02241 00 273 15

101325

Pa m mol K

3

PPa m mol K

3

Pa mmol K

P

3

0 02241 00 273 15

8 309

8 31

.. .

.

.

R

R

= ⋅ ⋅= aa m

mol K3

SDs allowed)⋅ ⋅ (3

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1 Measuring and Calculating

ls = 8.4 m, ws = 0.76 mlt = 189.2 cm, wt = 91.4 cm

Formulas: The formula for determining the area of a rectangular re-gion is A = l × w. Let the area of the student tabletop beAs = ls × ws = (lw)sand the area of the teacher’s demonstration desktop beAt = lt × wt = (lw)t.The total area is equal to the sum of the two individual areas, orAtotal = As + At.Combining these formulas, we obtainAtotal = (lw)s + (lw)t.Convert the teacher’s desktop dimensions to meters.

Substitute the measured values into the total area equation.Atotal = (8.4 m)(0.76 m) + (1.892 m)(0.914 m)Atotal = 6.38 m2 + 1.729 m2 (intermediate results)Note that the positions of the least significant digits in these prod-ucts were determined by Math Rule 3.

(allowed precision 0.1 m2, 2 SDs)

The SDs in the sum of the intermediate quantities were determined by Math Rule 2.

l

w

t

t

cm 1 m

100 cm m

cm 1 m

cm m

= =

= =

189 21 892

91 4

1000 914

..

..

6 38

8 109

.

.

m

1.729 m

m 8.1 m

2

2

2 2=

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2 Chapter Three

Note in the sample problem that you can use distinctive subscripts to uniquely identify similar variables in a formula. Here we used the subscript “s” to denote variables associated with the students’ table and “t” for those pertaining to the teacher’s desk.