Dimensional Analysis Step by Step (Unit Conversions)

Every equality between units gives you two conversion factors. Since $1 \text{ km} = 1000 \text{ m}$:

$$\frac{1000 \text{ m}}{1 \text{ km}} = 1 \qquad \frac{1 \text{ km}}{1000 \text{ m}} = 1$$

Multiplying by 1 does not change a quantity, only its units. The rule for choosing which version to use:

The unit you want to get rid of goes on the bottom. The unit you want goes on top.

Example: Convert 3.45 km to meters.

$$3.45 \text{ km} \times \frac{1000 \text{ m}}{1 \text{ km}} = 3450 \text{ m}$$

The km on top cancels the km on the bottom, leaving meters.

1. Write the starting quantity with its units.
2. Write the target unit so you know where you are going.
3. Plan a path of known equalities from the start unit to the target unit.
4. Set up the fractions so each unwanted unit cancels.
5. Cross out the units, then multiply the numbers on top and divide by the numbers on the bottom.
6. Check that only the target unit is left and that the size of the answer makes sense.

Example: Convert 2.50 hours to seconds.

Path: hours to minutes to seconds.

$$2.50 \text{ h} \times \frac{60 \text{ min}}{1 \text{ h}} \times \frac{60 \text{ s}}{1 \text{ min}} = 9000 \text{ s} = 9.00 \times 10^3 \text{ s}$$

Scientific notation shows the three significant figures from 2.50.

For a rate like miles per hour, convert the top unit and the bottom unit separately in the same chain.

Example: Convert 65 mi/h to m/s. Use $1 \text{ mi} = 1609.344 \text{ m}$ and $1 \text{ h} = 3600 \text{ s}$.

$$\frac{65 \text{ mi}}{1 \text{ h}} \times \frac{1609.344 \text{ m}}{1 \text{ mi}} \times \frac{1 \text{ h}}{3600 \text{ s}} = 29 \text{ m/s}$$

1. Miles is on top in the starting value, so the mile factor has miles on the bottom.
2. Hours is on the bottom in the starting value, so the hour factor has hours on top.
3. Numbers: $65 \times 1609.344 \div 3600 = 29.06$, which rounds to 29 m/s (two significant figures, from 65).

When a unit is squared or cubed, the conversion factor must be squared or cubed too.

Example 1: Convert 2.50 $\text{m}^2$ to $\text{cm}^2$.

$$2.50 \text{ m}^2 \times \left(\frac{100 \text{ cm}}{1 \text{ m}}\right)^2 = 2.50 \times 10\,000 \text{ cm}^2 = 2.50 \times 10^4 \text{ cm}^2$$

Using 100 instead of $100^2$ would give 250 $\text{cm}^2$, which is far too small. A square meter is 100 cm by 100 cm.

Example 2: Convert 0.750 L to $\text{cm}^3$.

Since $1 \text{ mL} = 1 \text{ cm}^3$ exactly:

$$0.750 \text{ L} \times \frac{1000 \text{ mL}}{1 \text{ L}} \times \frac{1 \text{ cm}^3}{1 \text{ mL}} = 750. \text{ cm}^3$$

Dimensional analysis is the backbone of chemistry math. Density, molar mass, and Avogadro's number are all conversion factors.

Example 1: Ethanol has a density of 0.789 g/mL. What is the mass of 250. mL?

$$250. \text{ mL} \times \frac{0.789 \text{ g}}{1 \text{ mL}} = 197 \text{ g}$$

Example 2: How many water molecules are in 12.0 g of water? Use 18.02 g/mol and $6.022 \times 10^{23}$ molecules/mol.

$$12.0 \text{ g} \times \frac{1 \text{ mol}}{18.02 \text{ g}} \times \frac{6.022 \times 10^{23} \text{ molecules}}{1 \text{ mol}} = 4.01 \times 10^{23} \text{ molecules}$$

1. Grams cancel, leaving moles: $12.0 \div 18.02 = 0.6659$ mol.
2. Moles cancel, leaving molecules: $0.6659 \times 6.022 \times 10^{23} = 4.01 \times 10^{23}$.

Significant figures: exact definitions like 1 km = 1000 m or 60 s = 1 min have unlimited significant figures and never limit your answer. Measured values like 0.789 g/mL do. Paste it into the VeilGPT solver to check your steps.