Question

What mass of lead (density 11.4 \(\mathrm{g} / \mathrm{cm}^{3}\) ) would have a volume identical to 15.0 \(\mathrm{g}\) of mercury (density 13.6 \(\mathrm{g} / \mathrm{cm}^{3} ) ?\)

Short answer

The mass of lead required to occupy the same volume as the mercury would be approximately \( (\frac{15.0}{13.6}) * (11.4) \) g.

Step-by-step solution

Find the volume of mercury

To find the volume of mercury, we will use the following formula: Volume = Mass / Density Given: Mass of mercury = 15.0 g Density of mercury = 13.6 g/cm³ Volume = \( \frac{15.0}{13.6} \)

Calculate the mass of lead

Now that we have found the volume of mercury, we will use it to find the mass of lead that would occupy the same volume using the density of lead. Given: Density of lead = 11.4 g/cm³ Volume of lead = Volume of mercury (from step 1) Mass of lead = Volume * Density Mass of lead = \( (\frac{15.0}{13.6}) * (11.4) \) The mass of lead required to occupy the same volume as the mercury would be approximately \( (\frac{15.0}{13.6}) * (11.4) \) g.

Key concepts

Mass and Volume Relationship

Understanding the relationship between mass and volume is crucial in density calculations. Density is defined as the mass of a substance per unit volume, expressed in the units of g/cm³ in this context. The formula for density is given by:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]From this equation, you can determine that mass is the product of density and volume:\[ \text{Mass} = \text{Density} \times \text{Volume} \]This shows that if you know the density of a material and the volume it occupies, you can calculate its mass. Conversely, if you have the mass and the density, you can find the volume, which is crucial in problems like this one.

• Density helps to convert between mass and volume: For a given volume, higher density means more mass and vice versa.
• Units are important: Always ensure your units for mass and volume match those used in the density calculation.

Using these relationships, problems involving different materials, like lead and mercury, can be solved efficiently.

Lead Properties

Lead is a dense metal, characterized by its relatively high density of 11.4 g/cm³. This property makes lead useful in situations where weight is an asset, such as in radiation shielding or weight balancing.

• High density: Lead's density is higher than many common materials, meaning a small volume of lead has a high mass.
• Usage: Lead is used in various industrial applications like batteries, weights, and protective equipment due to its density and malleability.

For the exercise, understanding lead's density is essential in calculating the mass needed to match a specific volume. In our situation, we used the density of 11.4 g/cm³ to find out how much mass lead would have if it occupied the same volume as mercury. This property helps distinguish it from other metals with a lower density, leading to different calculated masses for the same volume.

Mercury Properties

Mercury is unique for being the only metal that is liquid at room temperature, and it has a high density of 13.6 g/cm³. This makes mercury heavier than many substances with the same volume.

• Liquid state: Unlike most metals, mercury is liquid at room temperature, affecting how it's stored and used.
• Density comparison: With its density, mercury weighs significantly more than many other materials for the same volume.

In the exercise, knowing mercury's density was key to first determining its volume for a given mass. After finding out that volume, we then used similar concepts to solve for the lead's mass, by ensuring that both materials occupied the same volume. This process highlights the significant differences in density properties, affecting the resulting calculations for mass and volume in practical applications.