Density
Think of it this way: A brick and a sponge might be the same size (volume), but the brick feels heavier because its particles are packed more closely together. Therefore, the brick has a higher density.
Building on the concept of mass (the amount of matter) and volume (the space occupied), density is the ratio between them.
The Density Equation
The relationship is defined by:
\rho = \frac{m}{V}
Where:
- \rho (rho) is the density, measured in kilograms per cubic metre (kg/m^3) or grams per cubic centimetre (g/cm^3).
- m is the mass, measured in kilograms (kg) or grams (g).
- V is the volume, measured in cubic metres (m^3) or cubic centimetres (cm^3).
Rearranging the formula:
You can rearrange this equation to find mass or volume if you know the other two:
- To find mass: m = \rho \times V
- To find volume: V = \frac{m}{\rho}
This means density is the mass of a substance divided by the volume it occupies. The phrase "per unit" indicates division.
To find the density of a regularly shaped solid (e.g., a cube, rectangular block, or cylinder), you follow these steps:
- Measure Mass: Use an electronic balance to measure the mass (m) of the object in grams (g).
- Measure Dimensions: Use a ruler (for blocks) or calipers (for cylinders/spheres) to measure the necessary dimensions (length, width, height, or diameter).
- Calculate Volume: Use the appropriate geometric formula to calculate the volume (V). For example:
- Rectangular block: V = \text{length} \times \text{width} \times \text{height}
- Calculate Density: Substitute m and V into the density equation \rho = m/V.
For an irregularly shaped solid that sinks in water (e.g., a stone or key), you cannot use geometric formulas. Instead, you use the water displacement method.
- Measure Mass: Use a balance to find the mass (m) of the object.
- Initial Volume (V_1): Partially fill a measuring cylinder with water. Read the volume at the bottom of the meniscus (the curved surface of the liquid). This is V_1.
- Submerge Object: Gently lower the object into the water so it is fully submerged. Ensure no water splashes out.
- Final Volume (V_2): Read the new water level. This is V_2.
- Calculate Volume of Object: The volume of the object is the difference:
V_{\text{object}} = V_2 - V_1 - Calculate Density: Use \rho = m / (V_2 - V_1).
You cannot place a liquid directly on a balance pan. You must use a container (beaker or measuring cylinder).
- Measure Empty Mass: Measure the mass of an empty, dry beaker (m_{\text{empty}}) using a balance.
- Add Liquid: Pour a known volume of liquid into the beaker. You can measure this volume using a measuring cylinder before pouring, or by reading the level in the beaker if it is graduated.
- Measure Total Mass: Measure the mass of the beaker with the liquid (m_{\text{total}}).
- Calculate Mass of Liquid:
m_{\text{liquid}} = m_{\text{total}} - m_{\text{empty}} - Calculate Density: Use \rho = m_{\text{liquid}} / V_{\text{liquid}}.
Whether an object floats or sinks depends on the comparison between its density and the density of the fluid it is in.
- Floats: If \rho_{\text{object}} < \rho_{\text{fluid}}, the object will float.
- Sinks: If \rho_{\text{object}} > \rho_{\text{fluid}}, the object will sink.
- Neutrally Buoyant: If \rho_{\text{object}} = \rho_{\text{fluid}}, the object will remain suspended in the fluid.
Note: This topic is for the Supplement syllabus and is examined in Paper 2 and Paper 4.
When two liquids do not mix (are immiscible), they form layers based on their densities.
- The liquid with the lower density will float on top of the liquid with the higher density.
- Example: Oil (\rho \approx 0.92 g/cm^3) floats on water (\rho = 1.0 g/cm^3) because 0.92 < 1.0.
Step 1: Convert units.
The question asks for g/cm^3, so convert mass to grams:
m = 2.7 \text{ kg} = 2700 \text{ g}
Step 2: Calculate Volume.
V = 10 \times 5 \times 2 = 100 \text{ cm}^3
Step 3: Calculate Density.
\rho = \frac{m}{V} = \frac{2700}{100} = 27 \text{ g/cm}^3
The Correction: Remember the definition: Mass per unit Volume. The word "per" means division, and the quantity before "per" (mass) goes on top.
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
The Correction: The balance measures the mass of everything on it. You must subtract the mass of the empty container:
m_{\text{liquid}} = m_{\text{total}} - m_{\text{container}}
Why it matters: If you calculate density using kg and cm^3, your answer will be numerically incorrect for standard g/cm^3 expectations unless you convert. The standard conversion is:
1 \text{ g/cm}^3 = 1000 \text{ kg/m}^3
Example: If mass is in kg and volume in cm^3, convert mass to g first (\times 1000) or convert the final answer by multiplying by 1000.
Why it matters: Water forms a curve called a meniscus. Reading from the top of the curve causes parallax error and incorrect values.
Correct Usage: Always state that you read the volume at the bottom of the meniscus at eye level. This ensures accuracy.
Density = 45 / 15 = 3 g/cm³
- Measure mass of empty beaker.
- Pour liquid into beaker and measure total mass.
- Calculate mass of liquid (total - empty).
- Measure volume of liquid in cylinder.
- Calculate density = mass / volume.