Understanding Density, Mass, and Volume
Density, Mass, and Volume are key physical properties of matter. They are related by specific formulas.
- Mass (M): The amount of matter in an object. Common units include grams (g), kilograms (kg).
- Volume (V): The amount of space an object occupies. Common units include cubic centimetres (cm³), cubic metres (m³), millilitres (ml), litres (L).
- Density (D): The mass per unit volume of a substance. It tells us how tightly packed the matter is. Common units include grams per cubic centimetre (g/cm³), kilograms per cubic metre (kg/m³).
It's crucial to use consistent units when performing calculations. For example, if density is in g/cm³, mass should be in grams and volume in cm³.
Common Unit Conversions for Density Calculations
Being able to convert between units is essential.
Mass Conversions
1 kilogram (kg) = 1000 grams (g)1 gram (g) = 0.001 kilograms (kg)
Volume Conversions
1 litre (L) = 1000 millilitres (ml)1 millilitre (ml) = 1 cubic centimetre (cm³)1 litre (L) = 1000 cm³1 cubic metre (m³) = 1,000,000 cm³(100cm × 100cm × 100cm)1 cubic metre (m³) = 1000 litres (L)
The DMV Formula Triangle
A helpful way to remember the formulas is the DMV (or MDV) triangle:
To use the triangle:
- Cover the quantity you want to find.
- The remaining two quantities show you the formula:
- To find Mass (M), cover M: Density × Volume (D × V)
- To find Density (D), cover D: Mass ÷ Volume (M / V)
- To find Volume (V), cover V: Mass ÷ Density (M / D)
Mass = Density × Volume
Density = Mass ÷ Volume
Volume = Mass ÷ Density
Worked Examples
1. Calculating Density
Problem: An object has a mass of 150g and a volume of 50 cm³. What is its density?
Given: Mass = 150g, Volume = 50 cm³
Formula: Density = Mass ÷ Volume
Calculation: Density = 150g ÷ 50 cm³ = 3 g/cm³
Answer: The density is 3 g/cm³.
2. Calculating Mass
Problem: A substance has a density of 8 g/cm³. What is the mass of 20 cm³ of this substance?
Given: Density = 8 g/cm³, Volume = 20 cm³
Formula: Mass = Density × Volume
Calculation: Mass = 8 g/cm³ × 20 cm³ = 160 g
Answer: The mass is 160 g.
3. Calculating Volume
Problem: An object has a mass of 200g and a density of 4 g/cm³. What is its volume?
Given: Mass = 200g, Density = 4 g/cm³
Formula: Volume = Mass ÷ Density
Calculation: Volume = 200g ÷ 4 g/cm³ = 50 cm³
Answer: The volume is 50 cm³.
4. Unit Conversion Example
Problem: A block of metal has a mass of 2.5 kg and a volume of 500 cm³. Calculate its density in g/cm³.
Given: Mass = 2.5 kg, Volume = 500 cm³
Important: Units must be consistent for the density unit required (g/cm³). Mass needs to be in grams.
Workings for Mass Conversion:
- 1 kg = 1000 g.
- Convert 2.5 kg to grams:
2.5 kg × 1000 g/kg = 2500 g.
Formula: Density = Mass ÷ Volume
Calculation: Density = 2500 g ÷ 500 cm³ = 5 g/cm³
Answer: The density of the metal is 5 g/cm³.
Exam-Style Density, Mass & Volume Problems
1. Finding the Mass of Liquid
A container has a volume of 2 litres. It is filled with a liquid that has a density of 1.2 g/ml. What is the mass of the liquid in kilograms? (Remember 1L = 1000ml and 1ml = 1cm³).
Answer: 2.4 kg
Explanation:
- Convert volume to ml (or cm³):
2 litres × 1000 ml/litre = 2000 ml. - Given Density = 1.2 g/ml, Volume = 2000 ml.
- Formula: Mass = Density × Volume.
- Calculation: Mass =
1.2 g/ml × 2000 ml = 2400 g. - Convert mass to kilograms:
2400 g ÷ 1000 g/kg = 2.4 kg.
2. Identifying a Material
A piece of unknown material has a mass of 780g and a volume of 100 cm³. Material A has a density of 8.9 g/cm³. Material B has a density of 7.8 g/cm³. Material C has a density of 2.7 g/cm³. Which material is it likely to be?
Answer: Material B
Explanation:
- Calculate the density of the unknown material:
Density = Mass ÷ Volume =780g ÷ 100 cm³ = 7.8 g/cm³. - Compare this density to the given materials:
- Material A: 8.9 g/cm³
- Material B: 7.8 g/cm³
- Material C: 2.7 g/cm³
- The calculated density (7.8 g/cm³) matches Material B.
3. Comparing Population Densities
A student is researching population data. They use the following formula to calculate population density (K) in people per km², where P is the population in millions and M is the land area in million square miles:
K = P / (2.59 × M)
The student has the following data:
- Country X: Population of 143.96 million, Land Area of 6.593 million square miles.
- Country Y: Has a known population density of 3.57 people per km².
Answer: Country X has the higher population density.
Explanation:
- Calculate the population density for Country X:
- Use the given formula:
K = P / (2.59 × M). - Substitute the values for Country X:
K = 143.96 / (2.59 × 6.593). - First, calculate the denominator:
2.59 × 6.593 = 17.07587. - Now, calculate K:
K = 143.96 / 17.07587 ≈ 8.43 people per km². - Compare the densities:
- Country X Density: ~8.43 people per km²
- Country Y Density: 3.57 people per km²
- Since 8.43 > 3.57, Country X has a higher population density than Country Y.
Interactive DMV Calculator
Enter any two values to calculate the third. Important: Ensure your units are consistent for the two values you enter. The result will be in the corresponding unit (e.g., if mass is in 'g' and volume in 'cm³', density will be in 'g/cm³').
Calculated Value: ?
Key Points for Density, Mass & Volume
- Remember the MDV triangle (Mass at the top) to easily recall the formulas.
- Mass = Density × Volume
- Density = Mass ÷ Volume
- Volume = Mass ÷ Density
- Always ensure your units are consistent before calculating. Convert units if necessary.
- Density is a characteristic property of a substance.