Density and Experimental Methods for the ESAT

Defining Density

The density of a substance is defined as its mass per unit volume. While different samples of the same material may have different total masses or volumes, the density remains a constant characteristic property at a given temperature and pressure. This makes density a useful tool for identifying substances or distinguishing between different materials.

The relationship between mass, volume, and density is given by the formula:

$\text{density} = \frac{ ext{mass}}{\text{volume}}$

In symbolic form, this is written as:

$\rho = \frac{m}{V}$

Density is typically expressed in units of $kg\ m^{-3}$ or $g\ cm^{-3}$. For liquids, $g\ mL^{-1}$ is also frequently used.

Unit Conversions

It is common in ESAT questions to need to convert between $g\ cm^{-3}$ and $kg\ m^{-3}$. To perform these conversions, recall the following relationships:

$1\ \text{kg} = 10^3\ \text{g}$

$1\ \text{m}^3 = (100\ \text{cm})^3 = 10^6\ \text{cm}^3$

To convert from $g\ cm^{-3}$ to $kg\ m^{-3}$, you can use the conversion factor of $1000$. For example, the density of aluminium is $2.7\ g\ cm^{-3}$. To convert this to $kg\ m^{-3}$:

$\frac{2.7\text{g}}{1\ \text{cm}^3} = \frac{2.7 \times 10^{-3}\ \text{kg}}{1 \times 10^{-6}\ \text{m}^3} = 2.7 \times 10^3\ \text{kg m}^{-3} = 2700\ \text{kg m}^{-3}$

If an object is composed of multiple materials, it possesses an average density. This is calculated by dividing the total mass of the object by its total volume.

Worked Example: Aluminium Calculations

The density of solid aluminium is $2.700\ g\ cm^{-3}$.

a) Write the density of aluminium in $kg\ m^{-3}$. b) Calculate the volume of a $0.500\ kg$ sample of aluminium.

Solution for part a:

Since $1\ g\ cm^{-3} = 10^{-3}\ kg\ cm^{-3}$ and $1\ cm^{-3} = 10^6\ m^{-3}$:

$1\ g\ cm^{-3} = 10^{-3} \times 10^6\ kg\ m^{-3} = 1000\ kg\ m^{-3}$

Therefore, $2.700\ g\ cm^{-3} = 2.700 \times 1000 = 2700\ kg\ m^{-3}$

Solution for part b:

Rearrange the density formula to solve for volume: $V = \frac{m}{\rho}$

Ensure units are consistent. If using density in $g\ cm^{-3}$, convert mass to grams: $0.500\ kg = 500\ g$.

$V = \frac{500\ g}{2.7\ g\ cm^{-3}} = 185.18...\ cm^3$

Rounding to three significant figures gives $185\ cm^3$.

Experimental Determination of Densities

To determine density experimentally, you must measure both the mass and the volume of a sample. The specific method used depends on the state and shape of the sample.

1. Solid cuboids: Measure the mass using a balance. For volume, use a ruler or calliper to measure the length, width, and height. Multiply these dimensions together to find the volume.

2. Irregularly shaped solids: Measure the mass using a balance. To find the volume, use the displacement of water. Partially fill a measuring cylinder with water and record the initial volume. Submerge the sample completely. Record the new volume. The difference between the two readings is the volume of the solid. Note that the sample must be fully covered and must not cause the water to exceed the cylinder's scale.

3. Liquids: Place a measuring cylinder on a balance and record its mass (or tare the balance). Pour the liquid into the cylinder and record the new mass. Subtract the mass of the empty cylinder to find the mass of the liquid. The volume is read directly from the scale on the measuring cylinder.

Using a Displacement Can

For larger irregular solids, a displacement can (or eureka can) is used. The can is filled with water until it reaches the level of the spout. When the object is lowered into the water, it displaces a volume of water equal to its own volume. This overflow is collected in a measuring cylinder, providing a direct measurement of the sample's volume.

These displacement methods assume the solid does not dissolve in or react with the water, and that the solid sinks.

Objects that Float

If a solid is less dense than water, it will float. In this case, you must tie a heavy weight (a sinker) of known volume to the object. Submerge the combined pair and measure the total displaced volume. Subtract the volume of the sinker to find the volume of the original sample.

Worked Example: Floating Irregular Object

An irregularly shaped solid object has a mass of $35.2\ g$. A student ties a block of mass $447\ g$ and density $8.94\ g\ cm^{-3}$ to the object so that it sinks. The total volume of water displaced by the block and the object together is $105.0\ cm^3$. Calculate the density of the object.

Step 1: Find the volume of the block (sinker)

$V_{block} = \frac{m}{\rho} = \frac{447\ g}{8.94\ g\ cm^{-3}} = 50.0\ cm^3$

Step 2: Find the volume of the object

$V_{object} = V_{total} - V_{block} = 105.0\ cm^3 - 50.0\ cm^3 = 55.0\ cm^3$

Step 3: Calculate the density of the object

$\rho = \frac{m}{V} = \frac{35.2\ g}{55.0\ cm^3} = 0.640\ g\ cm^{-3}$

Comparing Densities of Solids, Liquids, and Gases

The particle model explains density variations through two factors: the mass of individual particles and the spacing between them.

Solids and Liquids

In most substances, solids are slightly denser than their liquid counterparts because particles are packed more closely in a regular arrangement. Generally, a solid is approximately 1.1 times denser than the liquid state of the same substance.

Water is a notable exception. Ice is less dense than liquid water because the molecules in ice are held in a rigid, open hexagonal lattice that keeps them further apart than the randomly arranged molecules in liquid water.

Gases

All substances are significantly less dense in the gaseous state. Particles in a gas are far apart, meaning there is much less mass per unit volume. A liquid's density typically decreases by a factor of roughly 1000 when it evaporates into a gas. Because gas particles are so spread out, their density is highly sensitive to temperature and pressure, which dictate particle separation.

Typical Values

• Solids and Liquids: Usually between $1000$ and $10,000\ kg\ m^{-3}$. Extremes include expanded polystyrene (about $20\ kg\ m^{-3}$) and gold (about $19,000\ kg\ m^{-3}$).
• Gases: Typically around $1\ kg\ m^{-3}$ at room temperature and atmospheric pressure.

It is possible for a specific solid to be less dense than a specific liquid. For instance, most woods float on water, and many solids float on liquid mercury. Some advanced materials like graphene aerogel are even less dense than air.

Worked Example: Particle Model Comparison

Use the particle model to explain why solids and liquids usually have similar densities but gases have much lower densities.

Solution:

According to the particle model, particles in both solids and liquids are in close contact, with very little empty space between them relative to their size. This results in a high mass per unit volume. In contrast, gas particles are separated by large distances. Consequently, there are far fewer particles in a given volume of gas, leading to a much lower mass per unit volume and a lower density.