Problem 32 - Chemistry: The Central Science

Question

A cylindrical rod formed from silicon is \(16.8 \mathrm{~cm}\) long and has a mass of \(2.17 \mathrm{~kg}\). The density of silicon is \(2.33 \mathrm{~g} / \mathrm{cm}^{3}\). What is the diameter of the cylinder? (The volume of a cylinder is given by \(\pi r^{2} h\), where \(r\) is the radius, and \(h\) is its length.)

Step-by-Step Solution

Answer

The diameter of the cylindrical silicon rod can be calculated using the given information and the following steps. 1. Convert the mass to grams: \(2.17 \mathrm{~kg} = 2170 \mathrm{~g}\). 2. Calculate the volume of the rod using the density: \(Volume = \dfrac{2170 \mathrm{~g}}{2.33 \mathrm{~g/cm^3}} \approx 931.33 \mathrm{~cm^3}\). 3. Calculate the radius of the rod using the volume and length: \(r = \sqrt{ \dfrac{931.33 \mathrm{~cm^3}}{ \pi \cdot 16.8 \mathrm{~cm}}} \approx 2.05 \mathrm{~cm}\). 4. Calculate the diameter of the rod: Diameter = 2 * r = \(2 \cdot 2.05 \mathrm{~cm} \approx 4.1 \mathrm{~cm}\).

Step 1: Convert mass to grams

To work with the given density, we need to convert the mass of the rod from kg to grams. 1 kg = 1000 grams

Step 2: Calculate the volume of the rod

Using the given density and mass of the rod, we can calculate the volume of the rod using the following formula: Volume = Mass / Density

Step 3: Calculate the radius of the rod

Now, using the cylindrical volume formula, we can solve for the radius: Volume = \(\pi r^2 h\) Where \(h\) is the length of the cylinder. Rearrange the formula to solve for \(r\): \(r = \sqrt{ \dfrac{Volume}{\pi h}}\)

Step 4: Calculate the diameter of the rod

Once we have found the radius, we can find the diameter by simply multiplying the radius by 2: Diameter = 2 * r

Key Concepts

Density

Density is a fundamental concept in physics and chemistry, which defines how much mass there is in a given volume. It's expressed as mass per unit volume, typically in units like grams per cubic centimeter (\(\text{g/cm}^3\)), kilograms per cubic meter (\(\text{kg/m}^3\)), or similar. Knowing the density of a material helps to understand how compact or spread out the mass is in a given space. To compute the density, you use the formula:- **Density = Mass / Volume**In this exercise, the density of silicon is provided as \(2.33 \, \text{g/cm}^3\). This means for every cubic centimeter of silicon, there is a mass of 2.33 grams. By using the density along with the mass, which must first be converted to grams to match the unit of density, you can then find specific geometric dimensions of objects made from silicon. Hence, understanding the density is crucial for calculating the volume when mass is known.

Mass Conversion

Mass conversion involves changing a mass from one unit to another, which allows consistency when carrying out calculations, especially when different units are involved. In the exercise, the mass of the silicon cylinder needs to be converted from kilograms to grams to use it with the density measure given in \(\text{g/cm}^3\). Here's a quick way to do it:- **1 kilogram equals 1000 grams**Thus, by multiplying the mass in kilograms by 1000, the mass is converted to grams. For example, a mass of \(2.17 \, \text{kg}\) becomes \(2.17 \, \text{kg} \times 1000 = 2170 \, \text{g}\). This straightforward conversion ensures that you can correctly use the value in further calculations without worrying about unit discrepancies.

Silicon Properties

Silicon is a semiconductor material widely used in electronics and engineering. Its unique properties make it a vital component. Some intrinsic characteristics include:- **Density:** As mentioned, silicon's density is \(2.33 \, \text{g/cm}^3\). - **Abundance:** Silicon is the second most abundant element in the Earth's crust. It is often found combined with oxygen in the form of silicon dioxide (SiO2).- **Electrical Properties:** Silicon is prized for its semiconductor properties, meaning it partly conducts electricity under certain conditions, essential for creating integrated circuits and microchips.Understanding these properties helps in engineering and scientific applications, making silicon an essential material in modern technology. When considering problems like finding the diameter of a silicon cylinder, knowing its density and how it behaves is instrumental in computing accurate results.