2.5: Converting Units-Dimensional Analysis

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Page ID: 521701

Sol Parajon Puenzo
Cañada College

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Learning Objectives

Convert a value reported in one unit to a corresponding value in a different unit.

The ability to convert from one unit to another is an important skill. For example, a nurse with 50 mg aspirin tablets who must administer 0.2 g of aspirin to a patient needs to know that 0.2 g equals 200 mg, so 4 tablets are needed. Fortunately, there is a simple way called dimensional analysis to convert from one unit to another. Dimensional analysis uses conversion factors.

Conversion Factors

Here is a simple example. How many centimeters are there in 3.55 m?

The relationship between cm and m is shown below:

\[100\; \rm{cm} = 1\; \rm{m} \nonumber \]

From this relationship, we can derive two conversion factors:

2 Conv factors.png

A fraction that has equivalent quantities in the numerator and the denominator but expressed in different units is called a conversion factor.

To solve the problem we use one of the two conversion factors. We first write the quantity we are given, 3.55 m. Then we multiply this quantity by conversion factor # 1.

\[ 3.55 \; \rm{m} \times \dfrac{100 \; \rm{cm}}{1\; \rm{m}} \nonumber \]

Conversion factor # 1 is chosen because m, the abbreviation for meters, occurs in both the numerator and the denominator of our expression, they cancel out. The final step is to perform the calculation that remains once the units have been canceled. We find that 3.55 m equals 355 cm.

conversion of meter to cm.

A generalized description of this process is as follows:

\[\text{quantity (in old units)} \times \text{conversion factor} = \text{quantity (in new units)} \nonumber \]

If you can master the technique of applying conversion factors, you will be able to solve a large variety of problems. Figure $\PageIndex{1}$ shows a concept map for constructing a proper conversion.

Concept map of conversions — Figure $\PageIndex{1}$: A Concept Map for Conversions. This is how you construct a conversion factor to convert from one unit to another.

Example $\PageIndex{1}$

Convert 81.2 g to kilograms following using the dimensional analysis. Show clearly conversion factors.

Solution

1 kg = 10³ g

Two conversion factors are possible:

and

Use the first conversion factor to cancel out the units of grams:

Significant Figures in Conversions

Numbers in conversion factors based on prefix changes, such as meters to centimeters, are not considered in the determination of significant figures in a calculation because the numbers in conversion factors are exact.

Exact numbers are defined or counted numbers, (not measured numbers), and can be considered as having an infinite number of significant figures. (In other words, 1 m is exactly 100 cm, by the definition of centi-.) Counted numbers are also exact. If there are 16 students in a classroom, the number 16 is exact. In contrast, conversion factors that come from measurements (such as density, as we will see shortly) are approximations and have a limited number of significant figures and should be considered in determining the significant figures of the final answer.

Example $\PageIndex{2}$

The average volume of blood in an adult male is 4.7 L. What is this volume in milliliters?
A hummingbird can flap its wings once in 18 ms. How many seconds are in 18 ms?

Solution

We start with what we are given, 4.7 L. We want to change the unit from liters to milliliters. There are 1,000 mL in 1 L. From this relationship, we can construct two conversion factors:

\[ \dfrac{1\; \rm{L}}{1,000\; \rm{mL}} \; \text{ or } \; \dfrac{1,000 \; \rm{mL}}{1\; \rm{L}} \nonumber \]

We use the conversion factor that will cancel out the original unit, liters, and introduce the unit we are converting to, which is milliliters. The conversion factor that does this is the one on the right.

$Liters converted to milliliters. A fraction shows 1000 milliliters on top, and 1 liter is on bottom.$

\[ 4.7 \cancel{\rm{L}} \times \dfrac{1,000 \; \rm{mL}}{1\; \cancel{\rm{L}}} = 4,700\; \rm{mL} \nonumber \]

Because the numbers in the conversion factor are exact, we do not consider them when determining the number of significant figures in the final answer. Thus, we report two significant figures in the final answer.

We can construct two conversion factors from the relationships between milliseconds and seconds:

\[ \dfrac{1,000 \; \rm{ms}}{1\; \rm{s}} \; \text{ or } \; \dfrac{1\; \rm{s}}{1,000 \; \rm{ms}} \nonumber \]

To convert 18 ms to seconds, we choose the conversion factor that will cancel out milliseconds and introduce seconds. The conversion factor on the right is the appropriate one. We set up the conversion as follows:

$Milliseconds converted to seconds. 1 second on top, and 1000 milliseconds on bottom of fraction.$

\[ 18 \; \cancel{\rm{ms}} \times \dfrac{1\; \rm{s}}{1,000 \; \cancel{\rm{ms}}} = 0.018\; \rm{s} \nonumber \]

The conversion factor’s numerical values do not affect our determination of the number of significant figures in the final answer.

Example $\PageIndex{3}$

Convert 35.9 kL to liters.
Convert 555 mm to meters.

Solution

We will use the fact that 1 kL = 1,000 L. Of the two conversion factors that can be defined, the one that will work is 1000L/ 1kL. Applying this conversion factor, we get:

\[35.9\, kL\times \dfrac{1000\,L}{1\,kL}= 35,900\, L \nonumber \nonumber \]

We will use the fact that 1,000 mm = 1 m. Of the two possible conversion factors, the appropriate one has the mm unit in the denominator:

\[\dfrac{1\,m}{1000\,mm} \nonumber \nonumber \]

Applying this conversion factor, we get:

\[555\,mm\times \dfrac{1m}{1000 mm}= 0.555\,m \nonumber \nonumber \]

Exercise $\PageIndex{1}$

Perform each conversion.

101,000 ns to seconds
32.08 kg to grams
607.8 kL ro mL
67.08 μL to liters
56.8 m to km

Answer

a. \[ 101,000 \cancel{\rm{ns}} \times \dfrac{1\; \rm{s}}{1,000,000,000\; \cancel{\rm{ns}}} = 0.000101\; \rm{s} \nonumber \]

b. \[ 32.08 \cancel{\rm{kg}} \times \dfrac{1,000 \; \rm{g}}{1\; \cancel{\rm{kg}}} = 32,080\; \rm{g} \nonumber \]

c. 6.078 × 10⁸ mL d. 6.708 × 10⁻⁵ L e. 5.68 × 10⁻² km

Exercise $\PageIndex{2}$

How do you determine which quantity in a conversion factor goes in the denominator of the fraction?
State the guidelines for determining significant figures when using a conversion factor.

Answer

The unit you want to cancel from the numerator goes in the denominator of the conversion factor.
Exact numbers that appear in many conversion factors do not affect the number of significant figures; otherwise, the normal rules of multiplication and division for significant figures apply.

Key Takeaway

Units can be converted to other units using the proper conversion factors.
Conversion factors are constructed from equalities that relate two different units.
Conversions can be a single step or multi-step.
Unit conversion is a powerful mathematical technique in chemistry that must be mastered.
Exact numbers do not affect the determination of significant figures.

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Example \(\PageIndex{1}\)

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Significant Figures in Conversions

Example \(\PageIndex{2}\)

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Example \(\PageIndex{3}\)

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Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)