Problem 36 - Chemistry The Central Science In Si Units

Question

(a) What is the mass of a silver cube whose edges measure 2.00 \(\mathrm{cm}\) each at \(25^{\circ} \mathrm{C} ?\) The density of silver is \(10.49 \mathrm{~g} / \mathrm{cm}^{3}\) at \(25^{\circ} \mathrm{C}\). (b) The density of aluminum is \(2.70 \mathrm{~g} / \mathrm{cm}^{3}\) at \(25^{\circ} \mathrm{C}\). What is the weight of the aluminum foil with an area of \(0.5 \mathrm{~m}^{2}\) and a thickness of \(0.5 \mathrm{~mm} ?\) (c) The density of hexane is \(0.655 \mathrm{~g} / \mathrm{mL}\) at \(25^{\circ} \mathrm{C} .\) Calculate the mass of \(1.5 \mathrm{~L}\) of hexane at this temperature.

Step-by-Step Solution

Answer

(a) 83.92 g, (b) 675 g, (c) 982.5 g.

Step 1: Calculate the Volume of the Silver Cube

To find the mass of the silver cube, first calculate its volume. Since it's a cube, use the formula for the volume of a cube: \[V = ext{side}^3\]Given that the edge of the cube measures 2.00 cm, the volume is:\[V = (2.00)^3 = 8.00 \text{ cm}^3\]

Step 2: Calculate the Mass of the Silver Cube

With the volume calculated, use the formula for mass, which is \[m = \text{density} \times \text{volume}\]Given the density of silver is 10.49 g/cm³, calculate the mass:\[m = 10.49 \times 8.00 = 83.92 \text{ g}\]

Step 3: Calculate the Volume of the Aluminum Foil

Convert the thickness from millimeters to centimeters since the density is given in terms of cm³: \[0.5 \text{ mm} = 0.05 \text{ cm}\]Now, calculate the volume using the formula for volume: \[V = \text{area} \times \text{thickness}\]The area is given in m², convert it to cm²:\[0.5 \text{ m}^2 = 5000 \text{ cm}^2\]Hence, the volume is:\[V = 5000 \times 0.05 = 250 \text{ cm}^3\]

Step 4: Calculate the Mass of the Aluminum Foil

Using the density of aluminum and the volume calculated, find the mass:\[m = \text{density} \times \text{volume} = 2.70 \times 250 = 675 \text{ g}\]

Step 5: Convert the Volume of Hexane to mL

Since the density of hexane is given in g/mL, convert 1.5 L to mL:\[1.5 \text{ L} = 1500 \text{ mL}\]

Step 6: Calculate the Mass of Hexane

Using the density of hexane to calculate the mass, use:\[m = \text{density} \times \text{volume} = 0.655 \times 1500 = 982.5 \text{ g}\]

Key Concepts

Silver Cube Mass

To calculate the mass of a silver cube with each edge measuring 2.00 cm, we must first determine its volume. A cube's volume can be calculated using the formula \( V = \text{side}^3 \). Since the cube's sides are 2.00 cm each, substituting into the formula gives us \( V = (2.00)^3 = 8.00 \text{ cm}^3 \).

Knowing the volume, we now calculate the mass using the density of silver, which is given as 10.49 g/cm³. The formula to find the mass from volume and density is \( m = \text{density} \times \text{volume} \). Therefore, substituting in the values, we find the mass as \( m = 10.49 \times 8.00 = 83.92 \text{ g} \).

By understanding how to compute volume and apply density, we can effectively determine the mass of any object given these parameters.

Aluminum Foil Weight

To calculate the weight of a piece of aluminum foil, we need to know its volume and density. The density of aluminum is 2.70 g/cm³, and the foil is given with an area of 0.5 m² and a thickness of 0.5 mm.

First, let's convert all our measurements to compatible units. We need to convert the thickness from millimeters to centimeters, resulting in 0.5 mm = 0.05 cm, and the area from square meters to square centimeters, so 0.5 m² = 5000 cm².

Then, calculate the volume with the formula \( V = \text{area} \times \text{thickness} \). Substituting in, \( V = 5000 \times 0.05 = 250 \text{ cm}^3 \).

Finally, calculate the weight (mass) of the aluminum using the formula \( m = \text{density} \times \text{volume} \). Substituting the known values, \( m = 2.70 \times 250 = 675 \text{ g} \). This result provides the weight of the aluminum foil effectively.

Hexane Mass Calculation

For calculating the mass of 1.5 L of hexane, first note its density: 0.655 g/mL. As density is provided in g/mL, you need the volume in milliliters.

Convert the given volume from liters to milliliters: \( 1.5 \text{ L} = 1500 \text{ mL} \).

Using the formula for mass \( m = \text{density} \times \text{volume} \), substitute these values into the equation: \( m = 0.655 \times 1500 \). This gives the hexane's mass as \( 982.5 \text{ g} \).

These calculations demonstrate how to use density and volume conversions to find the mass of any liquid, crucial for scientific and academic purposes.