Question

(a) A cube of osmium metal \(1.500 \mathrm{~cm}\) on a side has a mass of \(76.31 \mathrm{~g}\) at \(25^{\circ} \mathrm{C}\). What is its density in \(\mathrm{g} / \mathrm{cm}^{3}\) at this temperature? (b) The density of titanium metal is \(4.51 \mathrm{~g} / \mathrm{cm}^{3}\) at \(25^{\circ} \mathrm{C}\). What mass of titanium displaces \(125.0 \mathrm{~mL}\) of water at \(25^{\circ} \mathrm{C} ?\) (c) The density of benzene at \(15^{\circ} \mathrm{C}\) is \(0.8787 \mathrm{~g} / \mathrm{mL} .\) Calculate the mass of \(0.1500 \mathrm{~L}\) of benzene at this temperature.

Short answer

(a) The density of the osmium cube at $25^{\circ}\mathrm{C}$ is \(\frac{76.31\,\text{g}}{(1.500\,\text{cm})^3}\) = \(22.59\,\text{g/cm}^3\). (b) The mass of titanium that displaces $125.0\,\text{mL}$ of water at $25^{\circ}\mathrm{C}$ is (4.51 g/cm³) × (125.0 cm³) = \(564\,\text{g}\). (c) The mass of $0.1500\,\text{L}$ of benzene at $15^{\circ}\mathrm{C}$ is (0.8787 g/mL) × (0.1500 × 1000 mL) = \(131.8\,\text{g}\).

Step-by-step solution

Determine the volume of osmium cube

Given, the osmium cube has a side length of 1.500 cm. So we calculate the volume of the cube by using the formula for the volume of a cube: \(Volume = (side)^{3}\). Volume = \((1.500 \mathrm{~cm})^{3}\)

Calculate the density of the osmium cube

Density is defined as mass per unit volume: \(Density = \frac{mass}{volume}\) Given mass of the osmium cube = 76.31 g Volume of the osmium cube = \((1.500\,\text{cm})^3\) Density = \(\frac{76.31\,\text{g}}{(1.500\,\text{cm})^3}\) (b) To find the mass of titanium metal that displaces 125.0 mL of water:

Calculate the volume of titanium in cm³

We are given the volume of water displaced by the titanium, which is equal to the volume of the titanium itself. Volume of titanium = 125.0 mL (since 1 cm³ = 1 mL) Volume of titanium = 125.0 cm³

Calculate the mass of the titanium

Given, the density of titanium is 4.51 g/cm³ Density = \(\frac{mass}{volume}\) We need to find the mass. So, mass = Density × Volume Mass of Titanium = (4.51 g/cm³) × (125.0 cm³) (c) To find the mass of 0.1500 L of benzene at 15°C:

Convert the volume of benzene to mL

Given volume of benzene = 0.1500 L We know that 1 L = 1000 mL Volume of benzene = 0.1500 × 1000 mL

Calculate the mass of benzene

Given, the density of benzene at 15°C is 0.8787 g/mL Density = \(\frac{mass}{volume}\) We need to find the mass. So, mass = Density × Volume Mass of benzene = (0.8787 g/mL) × (0.1500 × 1000 mL)

Key concepts

Volume of a Cube

A cube is a special type of three-dimensional shape with all sides of equal length. To find the volume of a cube, we use a straightforward formula: \[ \text{Volume} = \text{side}^3 \] This formula means that you take the length of one side and multiply it by itself twice, which gives us the space inside the cube. Consider an example where the side of the cube is 1.500 cm. Plugging it into our formula: - Volume = \((1.500) \times (1.500) \times (1.500)\) cm³ - Volume = 3.375 cm³ This volume value is crucial, especially when you move on to calculating density since it represents the amount of space the cube occupies. Understanding this basic geometric concept helps you solve numerous problems involving volume measurements.

Mass Displacement

Mass displacement is a principle used to determine the mass of an object by measuring the volume of fluid it displaces. This concept is grounded in Archimedes' principle, which tells us that the volume of displaced fluid is equal to the volume of the object. In this exercise, we learn that if you immerse a titanium metal piece into water, the water level will increase. The volume of water displaced is the same as the volume of the titanium.

• Given: Volume of water displaced by titanium is 125.0 mL, which equals 125.0 cm³.
• To find the mass, use the density of the titanium, which is 4.51 g/cm³.
• Using the formula: \[ \text{Mass} = \text{Density} \times \text{Volume} \] means the mass is 4.51 g/cm³ \(\times\) 125.0 cm³.

This calculation requires the understanding that the properties of density link mass and volume directly.

Density Formula

Density is a fundamental property used to characterize substances, measured as mass per unit volume. The formula for density is written as: \[ \text{Density} = \frac{\text{mass}}{\text{volume}} \] This relation helps us find either the mass when the volume and density are known, or the volume when the mass and density are specified. Let's see this at work in the problem case with benzene: - Given: Density of benzene is 0.8787 g/mL. - We need to find the mass for 0.1500 L benzene. First, convert liters into milliliters to match the units of density (1 L = 1000 mL), so the volume is 150.0 mL. - Use formula: \[ \text{Mass} = \text{Density} \times \text{Volume} \] - Mass = \(0.8787\, \text{g/mL} \times 150.0 \,\text{mL}\) Through this, the interconnection between mass and volume is evident, facilitated by the density formula, imperative for many scientific and practical applications.