Problem 83 - Chemistry The Central Science In Si Units

Question

A thief plans to steal a cylindrical platinum medal with a radius of \(2.3 \mathrm{~cm}\) and a thickness of \(0.8 \mathrm{~cm}\) from a jewellery store. If the platinum has a density of \(21.45 \mathrm{~g} / \mathrm{cm}^{3},\) what is the mass of the medal in \(\mathrm{kg}\) ?

Step-by-Step Solution

Answer

The mass of the platinum medal is approximately 0.285 kg.

Step 1: Calculate the Volume of the Cylinder

The formula for the volume of a cylinder is given by \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height (or thickness) of the cylinder. Substitute \( r = 2.3 \) cm and \( h = 0.8 \) cm into the formula:\[V = \pi (2.3)^2 (0.8)\]\[V = \pi \times 5.29 \times 0.8 = \pi \times 4.232\]\[V \approx 13.29 \, \text{cm}^3\] (using \( \pi \approx 3.1416 \)).

Step 2: Calculate the Mass using Density

The mass \( m \) of an object can be calculated using the formula \( m = \text{density} \times \text{volume} \). Given that the density of platinum is \( 21.45 \, \text{g/cm}^3 \), multiply this with the volume calculated in Step 1:\[m = 21.45 \times 13.29 \approx 285.06 \, \text{g}\]

Step 3: Convert the Mass to Kilograms

Since 1 kilogram is equal to 1000 grams, divide the mass in grams by 1000 to convert it to kilograms:\[m = \frac{285.06}{1000} \approx 0.285 \, \text{kg}\]

Key Concepts

Volume of a cylinder

The volume of a cylinder is calculated using a specific mathematical formula that revolves around the dimensions of the cylinder. Imagine a soda can; it’s also cylindrical, similar to the platinum medal described in the exercise. To find how much space the cylinder takes up, known as its volume, we use the formula:\[ V = \pi r^2 h \] Here, "\( r \)" is the radius—half the diameter of the circular top—and "\( h \)" is the cylinder's height or thickness. For our platinum medal, the radius \( r \) is \( 2.3 \) cm, and the thickness \( h \) is \( 0.8 \) cm. When you place these values into the formula:- First, calculate \( r^2 \): \( (2.3)^2 = 5.29 \) cm².- Then, multiply by the thickness: \( 5.29 \times 0.8 = 4.232 \) cm³.- Finally, multiply by \( \pi \) (approximately \( 3.1416 \)): \( 4.232 \times \pi \approx 13.29 \) cm³. This figure represents the total space occupied by the medal, known as the cylinder's volume.

Density calculations

Density is like the weight of an item for each cubic centimeter measurement of space it occupies. Think of it as a measure of how packed or heavy a substance is. To find the mass of an object like our platinum medal, you use the formula:\[ m = \text{density} \times \text{volume} \] The density of platinum is given as \( 21.45 \, \text{g/cm}^3 \). This tells us that every cube measuring 1 cm on all sides, made of platinum, would weigh precisely \( 21.45 \) grams. Using this information, you multiply the volume we calculated earlier (\( 13.29 \, \text{cm}^3 \)) by the density:- \( 21.45 \times 13.29 \approx 285.06 \) grams. This calculation gives the weight of the medal in grams, illustrating how closely packed the particles in platinum are, forming the heavy medal.

Unit conversion

Unit conversion is the process of changing measurements from one unit to another. It's quite handy, especially when you want your answers to match a preferred measurement system or unit. For instance, in the situation with the platinum medal, converting grams to kilograms is essential because many people understand weight more clearly in kilograms. Conversion between these two units is straightforward: - **1 kilogram equals 1000 grams.** Therefore, to transform the medal's mass from grams to kilograms, you divide the number of grams by 1000: \[ m = \frac{285.06}{1000} \approx 0.285 \, \text{kg} \] So, the medal weighs approximately \( 0.285 \) kilograms. By understanding this conversion, you can switch between different measurement units effortlessly, which is especially useful in sciences and everyday life.