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 \(25^{\circ} \mathrm{C} ?(\mathrm{c})\) 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 volume of the osmium cube is \(V = a^3 = (1.500 \ \mathrm{cm})^3 = 3.375 \ \mathrm{cm}^3\). Therefore, the density of osmium is \(Density = \frac{mass}{volume} = \frac{76.31 \ \mathrm{g}}{3.375 \ \mathrm{cm}^3} = 22.6 \ \mathrm{g/cm}^3\). (b) The mass of titanium displacing \(125.0 \ \mathrm{mL}\) of water is \(Mass = Density \times Volume = 4.51 \ \mathrm{g/cm}^3 \times 125.0 \ \mathrm{cm}^3 = 563.8 \ \mathrm{g}\). (c) The mass of \(0.1500 \ \mathrm{L}\) of benzene at \(15^{\circ} \mathrm{C}\) is \(Mass = Density \times Volume = 0.8787 \ \mathrm{g/mL} \times 150.0 \ \mathrm{mL} = 131.8 \ \mathrm{g}\).

Step-by-step solution

(a) Calculate the volume of the osmium cube

First, we need to find the volume of the cube of osmium. The formula for the volume of a cube is: \[V = a^3\] where \(a\) is the side length of the cube. In this case, \(a = 1.500 \ \text{cm}\).

(a) Calculate the density of osmium

Now that we have the volume, we can calculate the density of osmium using the formula: \[Density = \frac{mass}{volume}\] Given the mass (\(76.31 \ \text{g}\)) and the calculated volume, we can find the density of osmium.

(b) Find the mass of titanium displacing water

In this problem, we are given the density of titanium metal and the volume of water displaced. We can use the formula for density to find the mass of titanium that displaces the given volume of water: \[Density = \frac{mass}{volume}\] Rearranging the formula, we can solve for mass: \[Mass = Density \times Volume\]

(c) Calculate the volume of benzene

In this problem, we are given the density of benzene, and we are asked to find the mass of a certain volume of benzene. First, we need to convert the volume of benzene in liters to milliliters: \[0.1500 \ \text{L} = 150.0 \ \text{mL}\]

(c) Calculate the mass of benzene

Now, we can use the formula for density to calculate the mass of benzene: \[Density = \frac{mass}{volume}\] Rearranging the formula, we can solve for mass: \[Mass = Density \times Volume\] Using the given density of benzene and the calculated volume, we can find the mass of benzene at the given temperature of \(15^{\circ} \mathrm{C}\).

Key concepts

Mass Calculation

Mass calculation is a fundamental concept in density exercises, as it ties together the mass and volume of a substance to find important information. In our scenario, we focus on calculating the mass of titanium and benzene. To start, the formula we use is:\[Mass = Density \times Volume\]This tells us that if we know the density and the volume of a substance, we can find out its mass.
For example, if we have a volume of benzene given as 0.1500 liters, we first need to convert it to milliliters to make units consistent, which becomes 150.0 mL. Then, using the density of benzene (0.8787 g/mL), we can find its mass by multiplying the density by the volume. Therefore,

• Mass of benzene = 0.8787 g/mL \( \times \) 150.0 mL.

Similarly, for titanium, after rearranging the formula, we can also calculate the mass of titanium by using its given density and the volume of water it displaces.
Understanding this relationship helps in practical applications, like determining how much of a substance we need in a chemical reaction.

Volume Calculation

Volume calculation is key when dealing with solids, particularly when measuring their capacity to occupy space. In our exercise, we calculated the volume of a cube of osmium. The general formula for the volume of a cube is as follows:\[V = a^3\]where \( a \) is the length of a side of the cube. This formula is straightforward because all sides of a cube have equal lengths.
In our example, with each side measuring 1.500 cm, we compute:

• The volume, \( V = (1.500 \ ext{cm})^3 \)

Calculating volume is essential, as it allows us to further compute the density or even mass of substances. In many chemistry problems, understanding the space a substance occupies compared to its mass or density provides insights into its properties and how it might react with others.
Learning to calculate volume is not just crucial for solid objects but can be extended to finding the volumes of liquids and irregular objects using displacement methods.

Density Formula

The density formula is central to solving problems that require understanding how firmly packed particles are in a given volume. Density is defined as mass per unit volume and is expressed by the formula:\[Density = \frac{mass}{volume}\]This relationship shows us that density is directly proportional to mass and inversely proportional to volume. Put simply, if you have a larger mass within a fixed volume, the density increases.
For example, in the case of osmium, by finding its mass as 76.31 g and its calculated volume from the cube formula, we can determine its density by placing these values into the equation:

• Density = \( \frac{76.31 \ ext{g}}{volume \ of \ cube} \)

Using the density formula allows us to solve a variety of problems, including predicting how materials will interact based on how much space they occupy and their mass. This fundamental concept is ubiquitous in science, from determining material purity to measuring fluid concentrations, making it critical to understand thoroughly.