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 L of benzene at this temperature.

Short answer

a) The density of the osmium metal cube is approximately \(22.6\, \dfrac{g}{\text{cm}^3}\). b) The mass of titanium that displaces 125.0 mL of water is approximately 563.8 g. c) The mass of 0.1500 L of benzene at 15°C is approximately 131.81 g.

Step-by-step solution

(1) Calculate the density of the osmium metal cube

We are given the dimensions and mass of an osmium metal cube. We need to find its density. To do this, we will first find the volume of the cube and then use the formula for density. The volume of a cube can be calculated using the formula: Volume = Side^3 The side length of the cube is given as 1.500 cm. So, the volume of the cube is: Volume = (1.500 cm)^3 = \(1.500^3\) cm³ ≈ 3.375 cm³ Now, we can use the formula for density: Density = Mass / Volume We know the mass of the cube is 76.31 g. So, the density of the osmium metal cube is: Density = 76.31 g / 3.375 cm³ ≈ \(22.6\, \dfrac{g}{\text{cm}^3}\)

(2) Calculate the mass of titanium that displaces 125.0 mL of water

We are given the density of titanium metal as 4.51 g/cm³ at 25°C. We need to find the mass of titanium that displaces 125.0 mL of water. First, we need to convert mL to cm³ (1 mL = 1 cm³). So, the volume of water displaced is 125 cm³. We know the density of titanium is 4.51 g/cm³. To find the mass of titanium, we rearrange the formula for density: Mass = Density x Volume Now we can plug in the values: Mass = 4.51 g/cm³ × 125 cm³ ≈ \(563.8\, g\)

(3) Calculate the mass of 0.1500 L of benzene at 15°C

The density of benzene at 15°C is given as 0.8787 g/mL. We need to find the mass of 0.1500 L of benzene. First, we need to convert L to mL (1 L = 1000 mL). So, the volume of benzene is 150 mL. Now, we can use the formula for density: Density = Mass / Volume We can rearrange the formula to find the mass of benzene: Mass = Density x Volume Now we can plug in the values: Mass = 0.8787 g/mL × 150 mL ≈ \(131.81\, g\) In summary: a) The density of the osmium metal cube is approximately \(22.6\, \dfrac{g}{\text{cm}^3}\). b) The mass of titanium that displaces 125.0 mL of water is approximately 563.8 g. c) The mass of 0.1500 L of benzene at 15°C is approximately 131.81 g.

Key concepts

Osmium

Osmium is among the densest naturally occurring elements on Earth and has a distinctive bluish-white color. It is often used in applications that require materials with a high density, such as in electrical contacts and fountain pen nibs. In this context, we are calculating the density of an osmium metal cube. To find the density, we need to know both its mass and volume. For a cube, volume can be easily calculated as the cube of its side length. Once we have the volume, we divide the mass by the volume to find the density. This property is crucial in distinguishing materials and understanding their behavior in different applications.

Titanium

Titanium is renowned for its strength and low density relative to its tensile strength, making it an excellent choice for aerospace, medical implants, and other high-performance applications. Despite being strong, titanium is very lightweight compared to other metals like steel. When calculating the mass of titanium that displaces a certain volume of water, we need to use its density. Density tells us how much mass is contained within a unit volume of metal. By multiplying the density of titanium by the volume of water displaced, we can determine the mass of the titanium. This calculation demonstrates how materials can be compared based on how much space they occupy versus their mass.

Benzene

Benzene is a simple aromatic hydrocarbon with a characteristic sweet smell. It is widely used as a solvent in chemical and industrial applications. At a specific temperature, such as 15°C, benzene has a defined density which can be used to find the mass of a given volume. By knowing the density, we can calculate the mass by multiplying the volume of benzene (typically converted from liters to milliliters) by its density. This calculation is useful in contexts where precise measurements of chemical substances are necessary, such as in laboratory experiments or industrial processes.

Volume

The concept of volume is fundamental in understanding the properties of materials and substances in chemistry and physics. Volume measures the amount of space occupied by an object or substance. For regular geometric shapes, such as cubes, calculating volume involves using simple mathematical formulas. For cubes, the volume is calculated by raising the side length to the power of three. Volume is a critical factor in density calculations, as it directly affects how we perceive the mass of materials and the space they occupy. Accurate volume measurement is essential for converting between units and ensuring precise calculations.

Mass

Mass is a measure of the amount of matter in an object and is typically measured in grams or kilograms. It is an inherent property of physical objects, not affected by the object's environment or the forces acting upon it. When calculating mass to determine density or to understand how substances behave when submerged in other materials, it serves as a key variable. The mass determines how an object sprawls or compresses when it occupies a certain space, showing its relationship with volume through the concept of density. By accurately measuring mass, you can predict how different materials will behave in different scenarios, an essential capability in scientific and industrial applications.