Density & Dimensional Analysis

Density and dimensional analysis are not glamorous topics, but they are the backbone of MCAT quantitative reasoning. Density connects mass to volume. Dimensional analysis connects everything else. Together, they let you solve multi-step problems by chaining conversion factors until your units work out.

Density

Key density values for the MCAT:

• Water: 1.00 g/mL (at 4 degrees C, where water is densest)
• Ice: 0.917 g/mL (less dense than liquid water - this is why ice floats)
• Most organic liquids: 0.7-0.9 g/mL (less dense than water)
• Most metals: much greater than 1.0 g/mL

Density as a Conversion Factor

Density is a fraction (g/mL), so you can use it as a conversion factor in two directions:

• Mass to volume: divide by density (or multiply by 1/d)
• Volume to mass: multiply by density

Example: What is the mass of 250 mL of ethanol (d = 0.789 g/mL)?

• mass = V x d = 250 mL x 0.789 g/mL = 197 g

Example: What volume does 500 g of mercury occupy (d = 13.6 g/mL)?

• V = m / d = 500 g / 13.6 g/mL = 36.8 mL

Specific Gravity

Specific gravity is commonly used in clinical settings. Urine specific gravity (normally 1.005-1.030) indicates how concentrated the urine is. Blood has a specific gravity of about 1.06. These values appear in MCAT passages.

Dimensional Analysis (Factor-Label Method)

Dimensional analysis is a systematic method for converting between units by multiplying by conversion factors that equal 1.

The rules:

1. Write down your starting value with its units
2. Multiply by conversion factors arranged so that unwanted units cancel
3. Continue until you reach the desired units
4. If the units work out, the math is set up correctly

Worked Example - Multi-Step Conversion

Problem: Convert 5.0 kg of aluminum to the number of atoms. (Molar mass of Al = 27.0 g/mol)

Step 1: Start with given: 5.0 kg

Step 2: Chain conversion factors:

$5.0 \text{ kg} \times (1000 \text{ g} / 1 \text{ kg}) \times (1 \text{ mol} / 27.0 \text{ g}) \times (6.022 \times 10^{23} \text{ atoms} / 1 \text{ mol})$

Step 3: Cancel units and calculate:

• 5.0 x 1000 = 5000 g
• 5000 / 27.0 = 185.2 mol
• $185.2 \times 6.022 \times 10^{23} =$ $1.11 \times 10^{26}$ atoms

Every unit cancels except “atoms” - which is exactly what we wanted.

Common Conversion Factors for the MCAT

Combining Density with Dimensional Analysis

Many MCAT problems require chaining density with other conversions. Density acts as just another conversion factor in the chain.

Example: How many molecules of ethanol are in 100 mL of pure ethanol? (d = 0.789 g/mL, MW = 46.07 g/mol)

$100 \text{ mL} \times (0.789 \text{ g} / 1 \text{ mL}) \times (1 \text{ mol} / 46.07 \text{ g}) \times (6.022 \times 10^{23} \text{ molecules} / 1 \text{ mol}) =$ $1.03 \times 10^{24}$ molecules

Each step converts one unit to the next: mL -> g -> mol -> molecules.

Density of Solutions

For solutions, density connects the total mass of the solution to its volume. This lets you convert between concentration units.

Example: Converting molarity to molality using density.

A 2.00 M NaCl solution has a density of 1.08 g/mL. Find the molality.

1. Take 1.00 L of solution as a basis
2. Moles of NaCl = 2.00 mol
3. Mass of solution = 1000 mL x 1.08 g/mL = 1080 g
4. Mass of NaCl = 2.00 mol x 58.44 g/mol = 116.9 g
5. Mass of solvent = 1080 - 116.9 = 963.1 g = 0.9631 kg
6. Molality = 2.00 mol / 0.9631 kg = 2.08 m

This type of conversion requires density as the bridge between volume-based (molarity) and mass-based (molality) concentration units.