Density and Specific Gravity Calculations — Formula and Worked Examples
Density looks like a Class 6 topic, and then it reappears in Class 12 as the step that turns a reagent bottle label into a molarity. Every bottle of concentrated hydrochloric acid in a school laboratory is labelled with a percentage and a density, and you cannot prepare a standard solution from it without combining the two. This guide covers the definitions, the unit conversions that catch people out, and four worked calculations including the label-to-molarity conversion.
Density
| Symbol | Meaning | Common units |
|---|---|---|
| ρ (rho) | density | g/cm³, g/mL, kg/m³, g/L |
| m | mass | g or kg |
| V | volume | cm³, mL, L or m³ |
Because 1 cm³ is exactly 1 mL, g/cm³ and g/mL are the same unit written two ways. Chemists usually write g/mL for liquids and g/cm³ for solids, but nothing changes.
Specific gravity (relative density)
For solids and liquids the reference is water, taken as 1.000 g/cm³. For gases the reference is usually air or hydrogen. Because it is a ratio of two densities, specific gravity has no unit — that is the whole point of it, and it is the most commonly asked one-mark question on the topic.
Water is at its densest at 4 °C, where ρ = 0.99997 g/cm³, which is why that temperature is the standard reference. At 25 °C water is slightly lighter, 0.997 g/cm³. Careful work quotes specific gravity with both temperatures, written as 20/4 °C, meaning "the substance measured at 20 °C compared with water at 4 °C". For school problems, treating the reference as exactly 1.000 g/cm³ makes the specific gravity numerically equal to the density in g/cm³.
Unit conversions that must be automatic
| From | To | Multiply by | Example |
|---|---|---|---|
| g/cm³ | kg/m³ | 1000 | 1.30 g/cm³ = 1300 kg/m³ |
| g/mL | g/L | 1000 | 1.18 g/mL = 1180 g/L |
| kg/L | g/cm³ | 1 | 1.84 kg/L = 1.84 g/cm³ |
| g/L | kg/m³ | 1 | 1.25 g/L = 1.25 kg/m³ |
The 1000-fold jump between g/cm³ and kg/m³ is where most numerical answers go wrong in physics-flavoured questions. Sanity check: water is 1 g/cm³ and 1000 kg/m³; if your answer for a liquid comes out near 1000 g/cm³, you have converted the wrong way.
Worked example 1 — the basic calculation
Problem: 25.0 mL of a liquid has a mass of 32.5 g. Find its density in g/mL and kg/m³, and its specific gravity.
ρ = 32.5 ÷ 25.0 = 1.30 g/mL
In SI: 1.30 × 1000 = 1300 kg/m³
Specific gravity = 1.30 ÷ 1.000 = 1.30 (no unit)
The liquid is denser than water, so it would sink if the two did not mix.
Worked example 2 — density of an irregular solid by displacement
Problem: A piece of metal has a mass of 53.8 g. Dropped into a measuring cylinder, it raises the water level from 20.0 mL to 26.0 mL. Find its density and suggest what the metal might be.
Step 1 — volume by displacement. V = 26.0 − 20.0 = 6.0 mL
Step 2 — density. ρ = 53.8 ÷ 6.0 = 8.97 g/cm³
Copper has a density close to 8.96 g/cm³, so copper is a reasonable suggestion. As with molecular mass from vapour density, a single physical constant narrows the field rather than proving an identity — brass and nickel are also in this range.
Two practical points examiners look for: the metal must not dissolve in or react with the liquid, and it must sink completely, or the displaced volume is not the volume of the solid.
Worked example 3 — from a reagent bottle label to a molarity
This is the calculation that makes density matter in chemistry. A bottle label gives you a percentage by mass and a density; a titration needs a molarity. Here is the derivation, which is worth understanding rather than memorising.
Take exactly 1 litre (1000 mL) of the solution. Its mass is 1000 × ρ grams. If the solution is P% by mass, the mass of solute in it is 1000 ρ × (P ÷ 100) = 10 ρ P grams. Dividing by the molar mass gives the moles in one litre, which is the molarity:
Problem: Concentrated hydrochloric acid is 36.0% HCl by mass and has a density of 1.18 g/mL. Find its molarity.
Step 1 — molar mass. M(HCl) = 1.008 + 35.45 = 36.458 g/mol
Step 2 — the long way, to see what is happening. One litre of the acid
weighs 1000 × 1.18 = 1180 g.
Mass of HCl in it = 0.360 × 1180 = 424.8 g
Moles = 424.8 ÷ 36.458 = 11.65 mol
Molarity = 11.65 mol/L
Step 3 — the formula, as a cross-check.
(10 × 36.0 × 1.18) ÷ 36.458 = 424.8 ÷ 36.458 = 11.65 ✓ Identical, because the formula is
just the long way written compactly.
Now sulphuric acid. Concentrated H₂SO₄ is 98.0% by mass with a density of 1.84 g/mL.
M(H₂SO₄) = 2 × 1.008 + 32.06 + 4 × 15.999 = 2.016 + 32.06 + 63.996 = 98.07 g/mol
Molarity = (10 × 98.0 × 1.84) ÷ 98.07 = 1803.2 ÷ 98.07 = 18.4 mol/L
Roughly 18 molar — which is why a burette of dilute acid is prepared from a very small measured volume of this stock. Safety point that is also an exam point: always add the concentrated acid to water, never water to acid. The dilution is strongly exothermic, and water added on top of dense acid can boil and spit.
Worked example 4 — % w/w and % w/v are not the same thing
Problem: The hydrochloric acid above is 36.0% w/w. Express it as % w/v.
% w/v means grams of solute in 100 mL of solution.
100 mL of the acid weighs 100 × 1.18 = 118 g
Mass of HCl in it = 0.360 × 118 = 42.5 g
So the acid is 42.5% w/v
In general, % w/v = % w/w × ρ. The two are equal only when the density is 1.00 g/mL, which for dilute aqueous solutions is very nearly true — and that is exactly why the difference goes unnoticed until a concentrated solution appears in a question.
Where density is used as a measurement
Because density is easy to measure and often varies smoothly with composition, it is routinely used as an indirect measure of concentration. A hydrometer — a weighted float that sinks to a depth set by the liquid's density — reads specific gravity directly. It is used to check the electrolyte in a lead-acid battery, where the acid is consumed as the battery discharges, so the specific gravity falls; a fully charged cell is commonly quoted at around 1.26 to 1.28, but you should follow the manufacturer's own specification rather than a remembered figure. The same instrument checks milk and sugar solutions in food work.
Two other density facts worth carrying:
- Ice is about 0.917 g/cm³, less dense than liquid water, which is why it floats and why lakes freeze from the top down. Hydrogen bonding forces an open structure in the solid.
- Density falls as temperature rises for almost every substance, because the same mass occupies more volume. Water between 0 °C and 4 °C is the famous exception, and it is the reason the reference temperature is 4 °C.
Common mistakes that cost marks
- Giving specific gravity a unit. It is a ratio. Write 1.30, not 1.30 g/cm³.
- Mixing g/cm³ and kg/m³. The factor is 1000, and it goes up when moving from g/cm³ to kg/m³.
- Including the container's mass. Weigh the empty container first and subtract.
- Assuming the density of a solution equals that of water. True enough for very dilute solutions, badly wrong for concentrated acids and brines.
- Confusing % w/w with % w/v. Convert with the density, as in example 4, before using either in a molarity calculation.
- Using the volume of solvent instead of the volume of solution. Molarity is per litre of final solution, and mixing does not always give additive volumes.
- Quoting a density with no temperature in a practical write-up. Density is temperature dependent, so the value alone is incomplete.
Where this appears in exams
| Exam | Typical question |
|---|---|
| CBSE/ICSE Class 9–10 | Density by displacement; relative density definition and calculation |
| CBSE/ICSE Class 11–12 | Converting percentage strength and density into molarity, molality or normality |
| Class 11–12 practical | Preparing a standard solution from a concentrated reagent |
| JEE/NEET | Concentration interconversions; density of gases from PV = nRT |
| IIT-JAM / CUET-PG | Partial molar volumes; why volumes are not always additive on mixing |
Most density errors are unit errors, not chemistry errors. The Unit Converter handles density, mass and volume conversions in both directions — g/cm³ to kg/m³, mL to L, g to kg — so a factor of 1000 never quietly changes your answer.
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