Density of insoluble granules

Determining the density of small insoluble particles, such as sand grains or lead shots, requires a precise method. The density bottle is the tool commonly used to measure the density of such particles. A density bottle is equipped with a tight-fitting stopper that has a small hole to allow excess liquid to escape, ensuring accurate volume measurements. The bottle typically has a volume of 25 ml, 50 ml, or 100 ml and is frequently used to measure the relative density of two immiscible liquids.

Density bottle

Procedure for determining the density of sand using a density bottle

1. Weigh the empty density bottle. First, measure the mass of the empty density bottle, including its stopper, and record it as $m_0$.

2. Add sand and weigh again. Remove the stopper, add a small amount of sand into the bottle, replace the stopper, and then measure the mass of the bottle with the sand inside. Record this mass as $m_1$.

3. Calculate mass of sand. The mass of the sand is obtained by subtracting the mass of the empty bottle from the mass of the bottle containing the sand:

   $\text{Mass of sand} = m_1 - m_0$

4. Fill the bottle with water. Remove the stopper again and fill the bottle with water until it is completely full. Replace the stopper securely.

5. Weigh the bottle with water and sand. Measure the mass of the bottle containing both water and sand. Record this mass as $m_2$.

6. Calculate mass of water. The mass of the water is determined by subtracting the mass of the bottle with sand from the mass of the bottle with both sand and water:

   $\text{Mass of water} = m_2 - m_1$

7. Calculate the volume of water. Since the density of water is 1 g/cm³, the volume of water is numerically equal to the mass of the water in grams. Thus, the volume of water ($V_w$) is:

   $V_{w} = m_2 - m_1 \quad (\text{in cm}^3)$

8. Calculate the volume of sand. The total volume of the density bottle is equal to the combined volume of both the sand and water. Since the volume of water is now known, the volume of sand ($V_s$) can be determined by:

   $V_{s} = V_{\text{bottle}} - V_{\text{water}}$

9. Calculate the density of sand. Finally, use the mass of sand and its volume to calculate the density of the sand using the formula:

   $\rho_{\text{sand}} = \frac{m_1 - m_0}{V_{\text{bottle}} - (m_2 - m_1)}$

Example calculation

Suppose the following masses are recorded:

• $m_0$ = mass of the empty density bottle = 50 g
• $m_1$ = mass of the bottle with sand = 150 g
• $m_2$ = mass of the bottle with sand and water = 250 g
• $m_3$ = mass of the bottle with water only = 300 g

Step 1: calculate mass of sand

$\text{Mass of sand} = m_1 - m_0 = 150 \, \text{g} - 50 \, \text{g} = 100 \, \text{g}$

Step 2: calculate mass of water

$\text{Mass of water} = m_2 - m_1 = 250 \, \text{g} - 150 \, \text{g} = 100 \, \text{g}$

Step 3: calculate volume of water

Since the density of water is 1 g/cm³, the volume of water is numerically equal to the mass of the water:

$V_{\text{water}} = 100 \, \text{cm}^3$

Step 4: calculate volume of sand

If the volume of the density bottle is 250 cm³, the volume of sand is:

$V_{\text{sand}} = V_{\text{bottle}} - V_{\text{water}} = 250 \, \text{cm}^3 - 100 \, \text{cm}^3 = 150 \, \text{cm}^3$

Step 5: calculate density of sand

Finally, the density of the sand is:

$\rho_{\text{sand}} = \frac{100 \, \text{g}}{150 \, \text{cm}^3} = 0.67 \, \text{g/cm}^3$

Procedure for determining the density of lead shots using a density bottle

The process for measuring the density of lead shots is similar to that of sand:

1. Weigh the empty density bottle. Measure the mass of the empty density bottle, $m_0$.
2. Add lead shots and weigh. Place a known amount of lead shots in the bottle, replace the stopper, and measure the mass of the bottle with the lead shots, $m_1$.
3. Fill the bottle with water. Fill the bottle with water, replace the stopper, and measure the mass, $m_2$.
4. Weigh water only. Finally, fill the bottle with only water and measure the mass, $m_3$.

Example calculation for lead shots

• $m_0$ = 50 g
• $m_1$ = 150 g (mass of the bottle and lead shots)
• $m_2$ = 200 g (mass of the bottle, lead shots, and water)
• $m_3$ = 250 g (mass of the bottle filled with only water)

The formula for calculating the density of the lead shots is:

$\rho_{\text{lead}} = \frac{m_1 - m_0}{(m_3 - m_0) - (m_2 - m_1)}$