Question

An empty vial weighs \(55.32 \mathrm{~g}\). (a) If the vial weighs \(185.56 \mathrm{~g}\) when filled with liquid mercury \(\left(d=13.53 \mathrm{~g} / \mathrm{cm}^{3}\right)\) what volume of mercury is in the vial? (b) How much would the vial weigh if it were filled with the same volume of water \(\left(d=0.997 \mathrm{~g} / \mathrm{cm}^{3}\right.\) at \(\left.25^{\circ} \mathrm{C}\right) ?\)

Short answer

The volume of mercury is 9.63 cm³. The vial would weigh 64.92 g if filled with water.

Step-by-step solution

- Determine the Mass of Mercury

Subtract the mass of the empty vial from the mass of the vial filled with mercury. \[ m_{Hg} = 185.56 \text{ g} - 55.32 \text{ g} = 130.24 \text{ g} \]

- Calculate the Volume of Mercury

Use the formula for density to calculate the volume, \( V = \frac{m}{d} \). \[ V_{Hg} = \frac{130.24 \text{ g}}{13.53 \text{ g/cm}^3} = 9.63 \text{ cm}^3 \]

- Calculate the Mass of Water

Use the volume of the mercury to find the mass of the same volume of water, using the density of water. \[ m_{H_2O} = V \times d_{H_2O} = 9.63 \text{ cm}^3 \times 0.997 \text{ g/cm}^3 = 9.60 \text{ g} \]

- Determine the Total Mass of the Vial Filled with Water

Add the mass of the water to the mass of the empty vial. \[ m_{vial + H_2O} = 55.32 \text{ g} + 9.60 \text{ g} = 64.92 \text{ g} \]

Key concepts

Mass

Understanding mass is crucial for solving problems involving density calculations. Mass refers to the amount of matter in an object and is usually measured in grams (g) or kilograms (kg). In our exercise, we calculated the mass of the liquid mercury by subtracting the mass of the empty vial from the mass of the vial filled with mercury. This gave us a mass of 130.24 g for the mercury. Knowing the mass is the first step in finding the volume of the substance, which can then help us solve other related problems. Remember, mass can change if you add or remove material from an object, but it does not depend on the object’s location or the force of gravity acting on it.

Volume

Volume is the amount of space that an object or substance occupies. It is typically measured in cubic centimeters (\text{cm}^3) or liters (L). In our exercise, the volume of the mercury was found using the formula for density: \( V = \frac{m}{d} \). After determining the mass of the mercury, we used its known density to calculate the volume. Through this, the volume was determined to be 9.63 \text{cm}^3. Volume is essential when dealing with fluids as it allows for conversions between different materials using their respective densities.

Density

Density is a measure of how much mass is contained in a given volume. It is often expressed in grams per cubic centimeter (g/cm³). The formula to calculate density is \( d = \frac{m}{V} \), where \(m\) is mass and \(V\) is volume. In our exercise, we used the densities of mercury and water to solve for various quantities. Using the given density of mercury (13.53 g/cm³), we calculated the volume of mercury in the vial. Then, we used this volume and the density of water (0.997 g/cm³) to find the mass of the water that would fill the same volume. Understanding the concept of density allows you to compare different substances and understand how they will behave in different situations. For example, mercury is much denser than water, which is why the same volume of mercury has a much higher mass compared to water.