Problem 14 - Chemistry and Chemical Reactivity

Question

The cup is a volume measure widely used by cooks in the United States. One cup is equivalent to 237 mL. If 1 cup of olive oil has a mass of \(205 \mathrm{g}\), what is the density of the oil (in grams per cubic centimeter)?

Step-by-Step Solution

Answer

The density of olive oil is approximately 0.865 g/cm³.

Step 1: Understanding the Problem

The problem asks us to compute the density of olive oil given its mass and volume. We know the volume of 1 cup is 237 mL (which is also 237 cm³ since 1 mL = 1 cm³) and its mass is 205 g.

Step 2: Density Formula

Density is calculated using the formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \), where mass is given in grams and volume in cubic centimeters (cm³).

Step 3: Substitute Known Values

Substitute the given values into the density formula:\( \text{Density} = \frac{205 \, \text{g}}{237 \, \text{cm}^3} \).

Step 4: Calculate the Density

Perform the division to find the density: \( \text{Density} = \frac{205}{237} \approx 0.865 \, \text{g/cm}^3 \).

Step 5: Conclusion

Thus, the density of the olive oil is approximately 0.865 grams per cubic centimeter.

Key Concepts

Volume Conversion

Volume conversion is a crucial step in understanding and solving problems that involve measurements in different units. In this exercise, we deal with the conversion between cups and milliliters (mL), which also directly equates to cubic centimeters (cm³). This is because 1 mL equals 1 cm³; hence, converting from mL to cm³ does not change the numerical value, only the units differ.

• A practical example of volume conversion would be translating the volume of 1 cup as it's often used in recipes for culinary purposes, into the scientific units of milliliters or cubic centimeters which are used in chemistry and physics.
• In our problem, 1 cup is equivalent to 237 mL, making it 237 cm³.
• This is a straightforward conversion since both mL and cm³ are direct measures of volume and are interchangeable.

When solving problems, always ensure your volume measurements are in the appropriate unit before performing any further calculations. This avoids unnecessary errors and helps in maintaining consistency across your calculations.

Mass Measurement

Mass measurement refers to determining how much matter an object or substance contains. Mass is often measured in grams or kilograms in the metric system.

• For instance, in this exercise, the olive oil's mass is provided as 205 grams.
• Mass gives us an idea of how much of the substance we have, and is key in calculations involving density.

Knowing the mass is vital because it allows us to apply formulas like the density formula accurately. In scientific contexts, precise mass measurement is imperative, often involving specific tools like digital balances to avoid inaccuracies.

Density Formula

The density formula is the heart of this exercise, providing a means to calculate how compact or spread out a substance's molecules are.
Density can be expressed using the formula:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]where the mass is measured in grams and the volume in cubic centimeters (cm³).

• This formula essentially tells us how much mass is packed into a unit of volume, explaining the concept of density.
• In simpler terms, it shows how heavy a substance is for its size.
• In our olive oil example, inserting the mass and volume values into the formula gives: \( \text{Density} = \frac{205 \, \text{g}}{237 \, \text{cm}^3} \).

After performing the division, the result is approximately 0.865 g/cm³. This means that every cubic centimeter of the olive oil weighs about 0.865 grams. Understanding and applying the density formula allows us to compare and analyze different materials based on their weight and spatial occupation.