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.
Open the Unit Converter →Concentration interconversions are the part of the Solutions chapter that rewards drilling. ABC Chemistry runs Class 11–12 chemistry coaching at the Gurugram centre and online classes across India, and students in Delhi, Noida and Gurgaon who want one-to-one help at home can arrange it through delhihometutor.com. Course details: abcchemistry.in.