Question

A \(0.50-\mathrm{g}\) sample of vegetable oil is placed in a calorimeter. When the sample is burned, \(18.9 \mathrm{~kJ}\) are given off. What is the caloric value, in \(\mathrm{kcal} / \mathrm{g}\), of the oil?

Short answer

The caloric value of the oil is 9.03 kcal/g.

Step-by-step solution

Convert Energy to Kilocalories

First, convert the energy released from kilojoules to kilocalories using the conversion factor: 1 kcal = 4.184 kJ.So, \[ 18.9 \text{ kJ} \times \frac{1 \text{ kcal}}{4.184 \text{ kJ}} \ = 4.516 \text{ kcal} \] (rounding appropriately).

Determine Caloric Value per Gram

Next, calculate the caloric value per gram by dividing the energy produced in kilocalories by the mass of the oil sample in grams. \[ \text{Caloric value} = \frac{4.516 \text{ kcal}}{0.50 \text{ g}} = 9.032 \text{ kcal/g}. \] (rounding appropriately to a reasonable number of significant figures).

Key concepts

calorimetry

Calorimetry is the science of measuring the heat of chemical reactions or physical changes. It involves using a device called a calorimeter to determine the amount of energy released or absorbed during a process.

The key idea is to measure the temperature change of a known mass of water (or another substance) as the process occurs. The temperature change, along with the known specific heat capacity of the substance, allows us to calculate the heat involved.

In our exercise, we used calorimetry to determine the energy given off by burning a sample of vegetable oil. By measuring the heat released and using the calorimeter data, we can find the caloric value of the oil. This is a crucial step in many real-world applications, such as food science and nutrition, where understanding energy content is essential.

energy conversion

Energy conversion is the process of transforming energy from one form to another. In our exercise, we converted energy from kilojoules (kJ) to kilocalories (kcal).

Kilojoules and kilocalories are both units of energy, but they are used in different contexts. Kilojoules are part of the International System of Units (SI), while kilocalories are often used in nutrition and food energy.

We used the conversion factor: 1 kcal = 4.184 kJ. This allowed us to convert the energy released by burning the oil from kilojoules to kilocalories. Here’s how the conversion was done:

• Given: 18.9 kJ
• Conversion factor: 1 kcal = 4.184 kJ
• Calculation: 18.9 kJ × (1 kcal / 4.184 kJ) = 4.516 kcal

It’s essential to use the correct unit conversions to ensure accurate measurements and results in science and engineering.

significant figures

Significant figures are the digits in a number that carry meaning and contribute to its precision. There are rules for determining which digits are significant.

1. All non-zero digits are significant (e.g., 123 has 3 significant figures).
2. Any zeros between significant digits are also significant (e.g., 1002 has 4 significant figures).
3. Leading zeros are not significant (e.g., 0.0025 has 2 significant figures).
4. Trailing zeros are significant if there’s a decimal point (e.g., 2.340 has 4 significant figures).

In calculations, the precision of the result is determined by the number with the fewest significant figures. In our exercise:

• The mass of the oil (0.50 g) has 2 significant figures.
• The energy released (4.516 kcal) was rounded to 4.52 kcal for 3 significant figures.

Finally, the calculated caloric value per gram (9.032 kcal/g) should be rounded to the same number of significant figures as the least precise measurement in the calculation, resulting in:
9.0 kcal/g.
Understanding significant figures ensures that the precision of measurements is accurately reflected in the results.