Problem 88 - Chemistry: Atoms First

Question

Osmium is one of the densest elements known. What is its density if 2.72 g has a volume of \(0.121 \mathrm{cm}^{3} ?\)

Step-by-Step Solution

Answer

The density of osmium is approximately \(22.48 \text{ g/cm}^{3} \).

Step 1: Understand the concept of density

Density is defined as the mass per unit volume of an object. It can be calculated using the formula: density \( d = \frac{m}{V} \) where \( m \) is the mass and \( V \) is the volume.

Step 2: Identify the known values

From the problem, we know the mass \( m = 2.72 \text{ g} \) and the volume \( V = 0.121 \mathrm{cm}^{3} \).

Step 3: Insert the known values into the formula

Substitute the values of mass and volume into the density formula: \( d = \frac{2.72 \text{ g}}{0.121 \mathrm{cm}^{3}} \).

Step 4: Calculate the density

Perform the division to obtain the density of osmium: \( d \approx \frac{2.72}{0.121} \text{ g/cm}^{3} \).

Step 5: Report the final answer

Complete the calculation to find the density to an appropriate number of significant figures: \( d \approx 22.48 \text{ g/cm}^{3} \).

Key Concepts

Density Formula

Density is a fundamental concept in science, describing how much 'stuff' is packed into a certain space. To calculate it, the density formula \( d = \frac{m}{V} \) is essential, where \( d \) is density, \( m \) is mass, and \( V \) is volume.

Understanding this formula is crucial because it tells you how the mass of a substance is distributed within its volume. For example, osmium's density is high because a small volume contains a lot of mass. That's why, with a mass of 2.72 grams in a volume of just 0.121 cubic centimeters, osmium is considered one of the densest elements.

Mass and Volume

Mass and volume are the two critical components needed to calculate the density of an object. Mass is a measure of the amount of matter in an object and is typically measured in grams (g) or kilograms (kg). Volume is the amount of space that an object occupies and is commonly measured in cubic centimeters (\(\mathrm{cm}^{3}\)) or liters (L).

When working with these quantities, it's essential to use the correct units and to convert between units if necessary. For the osmium example, the mass provided is 2.72 g, and the volume is 0.121 \(\mathrm{cm}^{3}\), already tailored for direct use in the density formula without the need for conversion.

Significant Figures

Significant figures are a way of indicating the precision of a measurement. They include all the certain digits in a number plus one uncertain or estimated digit.

When calculating the density of osmium, we must consider significant figures. Since 2.72 has three significant figures and 0.121 has three as well, our calculation for density should also reflect three significant figures. This leads us to report the final density of osmium as approximately 22.5 g/cm³ instead of 22.48 g/cm³, after rounding to three significant digits. This ensures our answer's precision matches the precision of the given data.