The mass of sample X is 20.0g. It was placed in a graduated cylinder and the water level rose from A (14 mL) to B (19 mL). What is the density of sample X? Pay close attention to significant figures.
The Correct Answer and Explanation is:
Density=VolumeMass
Step 1: Given Values
- Mass of sample X = 20.0 g (3 significant figures)
- Initial water level (A) = 14 mL
- Final water level (B) = 19 mL
Step 2: Calculate Volume Displaced
This is the volume of the object, which equals the change in water level: Volume=19 mL−14 mL=5 mL\text{Volume} = 19 \, \text{mL} – 14 \, \text{mL} = 5 \, \text{mL}Volume=19mL−14mL=5mL
The values 14 and 19 both have 2 significant figures, so the result should have 2 significant figures: Volume=5.0 mL\text{Volume} = 5.0 \, \text{mL}Volume=5.0mL
Step 3: Calculate Density
Density=20.0 g5.0 mL=4.0 g/mL\text{Density} = \frac{20.0 \, \text{g}}{5.0 \, \text{mL}} = 4.0 \, \text{g/mL}Density=5.0mL20.0g=4.0g/mL
- Mass has 3 significant figures.
- Volume has 2 significant figures.
- The result must be rounded to the least number of significant figures, which is 2.
✅ Final Answer:
4.0 g/mL\boxed{4.0 \, \text{g/mL}}4.0g/mL
Explanation
Density is a physical property defined as the mass of a substance per unit volume. To determine the density of an irregularly shaped object like sample X, one common method is water displacement, as done here. By noting how much the water level in a graduated cylinder rises when the object is added, we calculate the object’s volume.
In this case, sample X was added to water in a graduated cylinder. The water level increased from 14 mL to 19 mL, so the object displaced 5.0 mL of water, indicating its volume is 5.0 mL. The object’s mass was given as 20.0 g. Using the formula: Density=MassVolume=20.0 g5.0 mL=4.0 g/mL\text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{20.0 \, \text{g}}{5.0 \, \text{mL}} = 4.0 \, \text{g/mL}Density=VolumeMass=5.0mL20.0g=4.0g/mL
We must be careful with significant figures, which reflect the precision of measurements. The mass (20.0 g) has three significant figures, while the volume (5.0 mL) has two, because both the initial and final water levels (14 and 19 mL) are measured to the nearest whole number. According to the rules of significant figures, the result should be reported with the same number of significant figures as the value with the fewest—here, two. Hence, the final density is 4.0 g/mL.
Accurate attention to significant figures ensures proper scientific communication, reflecting the true precision of the measurements and preventing overstatement of accuracy.
