Dimensional Analysis (Multi-Step Conversions)量纲分析（多步换算）

Big Idea · Scale, Proportion, and Quantity (SPQ)AP Science Practice 5 · Mathematical Routines~30 min

Learning goals

  • Use dimensional analysis to set up multi-step conversions, multiplying conversion factors so the units cancel one by one
  • Convert compound units (such as km/h and g/cm³) and area and volume units
  • Use measured quantities such as density as conversion factors, and report results with the right significant figures

Before you start

  • Measurement and units (1.1.4)
  • Significant figures in calculations (2.1.1)

More than one step

In lesson 1.1.4, one conversion factor was enough to turn liters into milliliters. Many problems take several steps. A road sign says 115 km/h; what is that in m/s? Both the length unit and the time unit have to change. Chemistry is full of chains like this: mass to moles to particles, or volume to mass to moles. This lesson gives you one method that handles every multi-step conversion without getting lost.

Dimensional analysis

Treat units like algebra symbols: they multiply, divide, and cancel. Every conversion factor equals 1, so multiplying by it changes the units without changing the amount. Multiplying a string of conversion factors so that the units cancel one by one, until only the unit you want is left, is called dimensional analysis量纲分析 (or the factor-label method).

Work through these steps:

  1. Write the given quantity with its number and units.
  2. Plan the route for the units, such as km/h → m/h → m/s.
  3. For each step, choose a conversion factor that puts the unit you want to cancel on the opposite side: one on top, one on the bottom.
  4. Check that the right units are left, then do the arithmetic and round to the correct significant figures.
115 kmh×1000 m1 km×1 h3600 s=31.9 ms115\ \frac{\cancel{\text{km}}}{\cancel{\text{h}}} \times \frac{1000\ \text{m}}{1\ \cancel{\text{km}}} \times \frac{1\ \cancel{\text{h}}}{3600\ \text{s}} = 31.9\ \frac{\text{m}}{\text{s}}

km/h is a compound unit, so both the km on top and the h on the bottom have to be converted. The relationships 1 km = 1000 m and 1 h = 3600 s are definitions, so they don’t limit significant figures, and the answer keeps the 3 in 115.

🔧 Math tool card · Units & ConversionsSkip it if you already know this

Which way up? Put the unit you want to cancel on the opposite side. If it’s on top in your quantity, it goes on the bottom of the factor; if it’s on the bottom (like the h in km/h), it goes on top.

Square or cube the factor for areas and volumes. Since 1 m = 100 cm,

1 m3=(100 cm)3=106 cm31\ \text{m}^3 = (100\ \text{cm})^3 = 10^6\ \text{cm}^3

not 100 cm³. Also, 1 mL = 1 cm³ and 1 L = 1000 mL = 1 dm³.

Which conversion factor turns 2.5 m³ into cm³?

Measured quantities as conversion factors

Any ratio between two quantities can serve as a conversion factor, not just relationships between units. The most common is density. Ethanol’s density is 0.789 g/mL, meaning 1 mL of ethanol has a mass of 0.789 g, which gives two factors:

0.789 g1 mLor1 mL0.789 g\frac{0.789\ \text{g}}{1\ \text{mL}} \quad \text{or} \quad \frac{1\ \text{mL}}{0.789\ \text{g}}

Use the first to go from volume to mass and the second to go from mass to volume. Unlike a definition, a density is measured: this one has 3 significant figures, and those limit the answer.

The same goes for US customary units. Only some of their relationships to metric units are exact: 1 in = 2.54 cm is exact by definition, while 1 lb = 453.6 g, 1 mi = 1.609 km, and 1 gal = 3.785 L are rounded to 4 significant figures. AP Chemistry uses SI units, but you’ll meet these in everyday problems.

Choose a problem, then pick cards from the bank to build a chain. If a card points the wrong way, click ⇅ to flip it:

Try it: pick conversion factors from the bank and line them up so the units cancel one by one

Conversion factors (click one to add it to the end of the chain):

Example 1: Converting a volume unit

A fish tank holds 0.250 m³. How many liters is that?

Step-by-step solution

  1. Plan the route. m³ → cm³ (= mL) → L, using 1 cm³ = 1 mL.

  2. Convert m³ to cm³. 1 m = 100 cm, and volume is length cubed, so the factor is cubed too: 1 m³ = 100³ cm³ = 10⁶ cm³:

    0.250 m3×106 cm31 m3=2.50×105 cm3\begin{aligned} & 0.250\ \text{m}^3 \times \frac{10^{6}\ \text{cm}^3}{1\ \text{m}^3} \\ &= 2.50 \times 10^{5}\ \text{cm}^3 \end{aligned}

  3. Convert mL to L. 2.50 × 10⁵ cm³ = 2.50 × 10⁵ mL:

    2.50×105 mL×1 L1000 mL=2.50×102 L\begin{aligned} & 2.50 \times 10^{5}\ \text{mL} \times \frac{1\ \text{L}}{1000\ \text{mL}} \\ &= \boxed{2.50 \times 10^{2}\ \text{L}} \end{aligned}

  4. Significant figures. Every relationship is a definition, so the answer keeps the 3 in 0.250. A trailing zero in “250 L” would be ambiguous, so use scientific notation.

  5. Check. 1 m³ is 1000 L, so 0.250 m³ is 250 L, which agrees.

  6. A common mistake. Multiplying by 100 only (not cubed), which gives 25 L.

Example 2: Converting with density

What is the mass, in kilograms, of 2.0 L of ethanol? (Ethanol’s density is 0.789 g/mL.)

Step-by-step solution

  1. Plan the route. L → mL → g → kg. The step mL → g uses the density as a conversion factor: 1 mL of ethanol has a mass of 0.789 g.

  2. Convert L to mL.

    2.0 L×1000 mL1 L=2.0×103 mL2.0\ \text{L} \times \frac{1000\ \text{mL}}{1\ \text{L}} = 2.0 \times 10^{3}\ \text{mL}

  3. Use the density to turn volume into mass. Write the density as g/mL so that mL cancels:

    2.0×103 mL×0.789 g1 mL=1578 g2.0 \times 10^{3}\ \text{mL} \times \frac{0.789\ \text{g}}{1\ \text{mL}} = 1578\ \text{g}

  4. Convert to kg and round. 1578 g = 1.578 kg. 2.0 has 2 significant figures and 0.789 has 3; the 1000s are definitions. Keep 2:

    m=1.6 kgm = \boxed{1.6\ \text{kg}}

  5. Check. Ethanol is less dense than water, and 2.0 L of water is about 2.0 kg, so a bit less, 1.6 kg, makes sense.

Practice: is the conversion factor the right way up?
  1. Converting 3.5 kg to g by multiplying by 1000 g / 1 kg
  2. Converting 250 mL to L by multiplying by 1000 mL / 1 L
  3. Converting 45 min to h by multiplying by 1 h / 60 min
  4. Converting 72 km/h to km/min by multiplying by 60 min / 1 h
  5. Converting 5.0 m to cm by multiplying by 1 m / 100 cm
  6. Converting 12 g/L to g/mL by multiplying by 1 L / 1000 mL
  7. Converting 2.0 m³ to cm³ by multiplying by (100 cm / 1 m)³
  8. Finding the volume of 50.0 g of ethanol (0.789 g/mL) by multiplying by 0.789 g / 1 mL
  9. Converting 3.0 h to s by multiplying by 3600 s / 1 h

Common mistakes

Mistake 1: Using a factor upside down. The number comes out wrong by a large factor, but a unit check catches it every time: the units pile up instead of canceling.

Mistake 2: “1 m³ = 100 cm³.” A volume unit is a length unit cubed, so the factor must be cubed too: 1 m³ = 10⁶ cm³.

Mistake 3: Converting only half of a compound unit. Changing km/h to m/s by converting km to m but forgetting to convert h to s leaves the answer off by a factor of 3600. Check every unit.

Quick check

How many cubic centimeters are in 2.5 m³?

A car travels at 72 km/h. What is its speed in m/s?

Summary

  • Dimensional analysis: multiply a chain of conversion factors, putting each unit you want to cancel once on top and once on the bottom, until only the unit you want is left. Check the units before the number.
  • Convert both the top and the bottom of compound units, and square or cube factors for areas and volumes: 1 m³ = 10⁶ cm³.
  • Measured quantities such as density can be conversion factors, and their significant figures limit the answer; defined relationships are exact.
On to AP (for later)

Almost every AP calculation can be written as a chain of conversion factors: mass to moles to particles (chapter 2.6), and solution concentrations and gas volumes work the same way. AP free-response questions require you to show your work, and writing every unit and canceling it makes your reasoning easy to follow for the grader, and for you.

Practice this chapter: 2.1 · Measurement and Calculation · Basic, Medium, and Hard levels

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