Question

For the masses and volumes indicated, calculate the density in grams per cubic centimeter. a. mass \(=452.1 \mathrm{g} ;\) volume \(=292 \mathrm{cm}^{3}\) b. mass \(=0.14 \mathrm{lb} ;\) volume \(=125 \mathrm{mL}\) c. mass \(=1.01 \mathrm{kg} ;\) volume \(=1000 \mathrm{cm}^{3}\) d. mass \(=225 \mathrm{mg} ;\) volume \(=2.51 \mathrm{mL}\)

Short answer

The densities of the four objects are: a. \(≈ 1.55\,\mathrm{g/cm}^{3}\) b. \(≈ 0.508\,\mathrm{g/cm}^{3}\) c. \(= 1.01\,\mathrm{g/cm}^{3}\) d. \(≈ 0.0896\,\mathrm{g/cm}^{3}\)

Step-by-step solution

Calculate the density for the first object (a)

Mass: 452.1 g; Volume: 292 cm³. Density of the object (a) = Mass / Volume \(Density = \frac{452.1\,\mathrm{g}}{292\,\mathrm{cm}^{3}}\) \(Density ≈ 1.55\,\mathrm{g/cm}^{3}\)

Calculate the density for the second object (b)

Mass: 0.14 lb; Volume: 125 mL. First, we convert mass and volume to the appropriate units. Mass conversion: 1 lb = 453.6 g Mass = 0.14 lb × 453.6 g/lb ≈ 63.5 g Volume conversion: 1 mL = 1 cm³ Volume = 125 mL × 1 cm³/mL = 125 cm³ Density of object (b) = Mass / Volume \(Density = \frac{63.5\,\mathrm{g}}{125\,\mathrm{cm}^{3}}\) \(Density ≈ 0.508\,\mathrm{g/cm}^{3}\)

Calculate the density for the third object (c)

Mass: 1.01 kg; Volume: 1000 cm³. First, we convert mass to the appropriate unit. Mass conversion: 1 kg = 1000 g Mass = 1.01 kg × 1000 g/kg = 1010 g Density of object (c) = Mass / Volume \(Density = \frac{1010\,\mathrm{g}}{1000\,\mathrm{cm}^{3}}\) \(Density = 1.01\,\mathrm{g/cm}^{3}\)

Calculate the density for the fourth object (d)

Mass: 225 mg; Volume: 2.51 mL. First, we convert mass and volume to the appropriate units. Mass conversion: 1 mg = 0.001 g Mass = 225 mg × 0.001 g/mg = 0.225 g Volume conversion: 1 mL = 1 cm³ Volume = 2.51 mL × 1 cm³/mL = 2.51 cm³ Density of object (d) = Mass / Volume \(Density = \frac{0.225\,\mathrm{g}}{2.51\,\mathrm{cm}^{3}}\) \(Density ≈ 0.0896\,\mathrm{g/cm}^{3}\) The densities of the four objects are: a. ≈ 1.55 g/cm³ b. ≈ 0.508 g/cm³ c. = 1.01 g/cm³ d. ≈ 0.0896 g/cm³

Key concepts

Unit Conversion in Chemistry

Understanding and performing unit conversions is a crucial skill in chemistry. In most reactions and calculations, different units such as kilograms, grams, pounds, milliliters, and cubic centimeters are involved. It ensures that measurements are consistent and comparable. Let's break down some common conversions.

• Mass Conversions: To convert from pounds to grams, you use the conversion factor 1 pound = 453.6 grams. For example, if you have 0.14 lb, you'd calculate: \[0.14 \, \text{lb} \times 453.6 \, \frac{\text{g}}{\text{lb}} = 63.5 \, \text{g}\]Converting kilograms to grams is simpler since 1 kg = 1000 grams. For conversion, simply multiply the weight in kilograms by 1000.
• Volume Conversions: Milliliters and cubic centimeters are often used interchangeably in chemistry because they are equivalent (1 mL = 1 cm³). Therefore, if you measure a volume as 125 mL, it stays 125 cm³ when performing calculations.

Making these conversions correctly is essential before using mass and volume in any of your calculations, such as in our density exercises.

Mass and Volume Relationship

The relationship between mass and volume is foundational in determining density, which is a basic property of matter. Density is defined as the mass of an object divided by its volume and is mathematically expressed as: \[ \text{Density} (\rho) = \frac{\text{Mass} (m)}{\text{Volume} (V)} \]This formula shows that density depends on two key factors — how much matter (mass) there is and the space it occupies (volume).Understanding the concept of density allows us to:

• Compare Substances: Different substances may have different densities. This helps in identifying materials based on how "heavy" they are for their size.
• Predict Behavior: Substance density can predict whether an object will float or sink in a fluid (e.g., a less dense object will float).

In the given exercise, densities were calculated for each object by converting all measurements to compatible units and then applying the density formula.

Problem-Solving in Chemistry

Chemistry often involves solving problems methodically and carefully. For density calculations, this involves multiple steps to ensure accuracy. Here's a simple approach:

• Identify and Understand: Recognize the given values, such as mass and volume, and the units used.
• Convert Units if Necessary: Especially in density-related problems, mass and volume need to be in grams and cubic centimeters, respectively.
• Apply the Correct Formula: Use the right formula to calculate density from the mass and volume.
• Verify and Analyze Results: After calculating, review the results for accuracy and reasonableness (e.g., comparing with expected density ranges for common substances).

These steps teach a structured approach to problem-solving in chemistry, hinging heavily on correct foundational knowledge like unit conversion and conceptual understanding of mass and volume relationships.