Problem 80 - Chemistry The Molecular Nature of Matter

Question

Titanium is a metal used to make golf clubs. A rectangular bar of this metal measuring \(1.84 \mathrm{~cm} \times 2.24 \mathrm{~cm} \times 2.44 \mathrm{~cm}\) was found to have a mass of \(45.7 \mathrm{~g}\). What is the density of titanium in \(\mathrm{g} \mathrm{cm}^{-3}\) ?

Step-by-Step Solution

Answer

\( \rho = \frac{45.7 \text{ g}}{1.84 \times 2.24 \times 2.44 \text{ cm}^3} = \frac{45.7 \text{ g}}{10.005504 \text{ cm}^3} \approx 4.57 \text{ g/cm}^3 \)

Step 1: Calculate the volume of the titanium bar

To calculate the volume of a rectangular bar, use the formula for the volume of a rectangle, which is the product of its length, width, and height. So, the volume V is given by \( V = \text{length} \times \text{width} \times \text{height} = 1.84 \text{ cm} \times 2.24 \text{ cm} \times 2.44 \text{ cm} \).

Step 2: Perform the multiplication to determine the volume

Multiplying the given dimensions, we get \( V = 1.84 \times 2.24 \times 2.44 \text{ cm}^3 \). Calculate this value to find the volume.

Step 3: Calculate the density of titanium

Density (\( \rho \)) is defined as the mass of an object divided by its volume. Using the formula \( \rho = \frac{\text{mass}}{\text{volume}} \), and the mass of the titanium bar, calculate the density of titanium.

Key Concepts

Chemical Properties

When we talk about chemical properties, we are referring to characteristics that allow a substance to undergo certain chemical changes or reactions because of its composition. For instance, titanium, which is often used to manufacture sporting goods like golf clubs, has a very high strength-to-weight ratio and excellent corrosion resistance. This is due to its chemical properties, which involve the arrangement of its atoms and its ability to form oxides that protect it from further corrosion.

Understanding the chemical properties can help us comprehend why a material like titanium is suitable for specific applications. It's these properties that contribute to the practical uses of the metal, such as in the aerospace, medical, and automotive industries.

As students progress in their study of materials, it's important to connect the chemical properties to real-world applications. This helps in grasping the importance of learning and understanding these properties beyond the classroom or textbook exercise.

Measurement Units

Measurement units are a fundamental part of scientific and mathematical problems, serving as the language that conveys quantity and allows for clear communication of results. In our exercise involving titanium, we use centimeters (cm) to measure length and grams (g) to measure mass. The density is then calculated and presented in grams per cubic centimeter (\(g/cm^3\)), which is a derived unit that combines mass and volume units.

Understanding and correctly using measurement units is critical for accuracy in scientific calculations. It ensures that the values calculated can be compared and reproduced. There's also a need to be capable of converting units where necessary, as standardized units of measurement facilitate shared understanding and are essential for international collaboration in science and engineering.

Material Density

Material density is a key concept in understanding how materials interact in different environments. It is calculated by dividing an object's mass by its volume (\( \rho = \frac{\text{mass}}{\text{volume}} \)). For instance, the density of the titanium used in our exercise is a reflection of how tightly its atoms are packed together. Knowledge of a material's density can lead to insights into its structural integrity and lightness, which correlates with its practical applications.

In the context of the exercise, the accurate calculation of titanium's density is important for quality control and for engineers to make informed decisions regarding its use in manufacturing. Making sure the calculation is comprehensible and correct allows us to anticipate the behavior of titanium in different conditions, such as when it is used to make golf clubs that might undergo substantial stress during use.