Problem 92 - Chemistry and Chemical Reactivity

Question

According to the Nutrient Data Laboratory website (www, ars. usda.gov/ba/bhnrc/ndl), corn oil contains \(3766 \mathrm{kJ}\) of energy per \(100 .\) g serving. (a) What is the energy content of \(100 .\) g of corn oil in units of nutritional calories (Cal)? (b) How many tablespoons of corn oil have an energy content equivalent to 1500 nutritional calories? (1.0 Tbsp \(=14\) g of corn oil) (c) What mass of water can be heated from \(25.0^{\circ} \mathrm{C}\) to its boiling point of \(100.0^{\circ}\) C using the energy of combustion of 1.00 Tbsp of corn oil?

Step-by-Step Solution

Answer

(a) 900 Cal (b) 12 tablespoons (c) 1676 g of water

Step 1: Convert energy from kJ to Cal

The energy content in corn oil is given as 3766 kJ per 100 g. To convert this to nutritional calories, we use the conversion factor: 1 kJ = 0.23900574 Cal.The calculation is as follows:\[3766 \text{ kJ} \times 0.23900574 \frac{\text{Cal}}{\text{kJ}} \approx 900 \text{ Cal}\]This means that 100 g of corn oil contains approximately 900 nutritional calories.

Step 2: Calculate energy content of 1 tablespoon of corn oil

1 tablespoon of corn oil has a mass of 14 g. We know from Step 1 that 100 g contains roughly 900 Cal.To find the energy content of 1 tablespoon:\[\frac{900 \text{ Cal}}{100 \text{ g}} \times 14 \text{ g} = 126 \text{ Cal}\]Therefore, 1 tablespoon of corn oil contains approximately 126 nutritional calories.

Step 3: Calculate tablespoons for 1500 nutritional calories

We need to find out how many tablespoons correspond to 1500 nutritional calories. Using the result from Step 2:\[1500 \text{ Cal} \div 126 \text{ Cal/Tbsp} \approx 11.9 \text{ Tbsp}\]Hence, approximately 12 tablespoons of corn oil would be needed to reach 1500 nutritional calories.

Step 4: Determine energy content of 1 tablespoon in kJ

We previously calculated in Step 2 that 1 tablespoon contains 126 Cal. Convert this back to kJ using the conversion factor: 1 Cal = 4.184 kJ.\[126 \text{ Cal} \times 4.184 \text{ kJ/Cal} \approx 527.2 \text{ kJ}\]So, 1 tablespoon contains approximately 527.2 kJ of energy.

Step 5: Calculate energy required to heat water

We know the energy content of 1 tablespoon of corn oil from Step 4 is about 527.2 kJ. We will use this energy to heat water from 25°C to 100°C. The formula for the energy required to heat water is:\[q = mc\Delta T\]where:- \( q \) is the heat energy in joules (J)- \( m \) is the mass of water in grams- \( c \) is the specific heat capacity of water (4.184 J/g°C)- \( \Delta T \) is the change in temperature (100°C - 25°C = 75°C)Convert 527.2 kJ to J:\[527.2 \text{ kJ} \times 1000 \frac{\text{J}}{\text{kJ}} = 527200 \text{ J}\]Use the energy formula to solve for \( m \):\[m = \frac{527200 \text{ J}}{4.184 \text{ J/g°C} \times 75 °C} \approx 1675.9 \text{ g}\]Thus, approximately 1676 g of water can be heated from 25°C to 100°C using the energy from 1 tablespoon of corn oil.

Key Concepts

Nutritional Calories

When we talk about nutritional calories, we're referring to the measurement of energy obtained from food. These are often displayed within food labels to help consumers understand how much energy they might gain from eating a certain food. Nutritional calories are technically kilocalories, which means each unit is equivalent to 1,000 small calories. However, for the sake of simplicity, 1 kilocalorie (kcal) is referred to as a "calorie" (Cal) on food labels.

When converting energy from joules or kilojoules, we should remember the conversion factor: 1 kilojoule (kJ) is approximately 0.239 nutritional calories (Cal). This conversion is crucial in translating energy values given in scientific contexts into practical information for nutritional labeling.

Specific Heat Capacity

Specific heat capacity is a property of a substance that tells us how much heat energy is needed to raise the temperature of a given quantity of the substance by a certain degree. For water, this value is notably high: 4.184 joules per gram per degree Celsius (J/g°C).

This means that water requires a lot of energy to change its temperature, which makes it a good stable environment for many biological processes. The high specific heat capacity of water also implies that it can absorb or release large amounts of heat with minimal changes in its own temperature. Understanding specific heat capacity is crucial for calculations involving heating substances, where it will define how the energy input translates to changes in temperature.

Energy Content Calculation

Calculating the energy content in foods or other materials involves converting values from one set of units to another, allowing for a better understanding or practical application. In our example, the energy content of corn oil is given in kilojoules. To find out how much energy it represents in terms of nutritional calories, we rely on conversion factors between units.

This process includes:

• Identifying the original units (e.g., kJ) and the desired units (e.g., Cal).
• Using the appropriate conversion factor. For example, 1 kJ = 0.239 Cal.
• Applying this factor to calculate the desired value in new units.

As a result, we can better understand food energy content and make accurate comparisons or dietary plans based on these calculations.

Temperature Change in Water

To comprehend how much energy is needed to heat water (or any substance), it’s essential to understand the terms involved in the formula: \( q = mc\Delta T \).

Where:

• \( q \) is the amount of energy in joules.
• \( m \) is the mass of the water in grams.
• \( c \) is the specific heat capacity (for water, \( 4.184 \text{ J/g°C} \)).
• \( \Delta T \) is the temperature change.

To compute the amount of energy required for a certain temperature change, we input these values into the formula, ensuring the energy input is matched with the mass and specific heat of the water.
This method can be applied to determine how much water can be heated, given a certain amount of energy, like from the combustion of a substance like corn oil.