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A 2 -in. piece of chocolate cake with frosting provides \(1670 \mathrm{kJ}\) of energy. What is this in dietary Calories (Cal)?

Short Answer

Expert verified
Approximately 399 Cal.

Step by step solution

01

Understand the Conversion Factor

Recognize that the conversion factor between kilojoules and dietary Calories is \( 1 \text{ Cal} = 4.184 \text{ kJ} \). This means that in order to convert energy in kilojoules to dietary Calories, you must divide the kilojoules by 4.184.
02

Perform the Calculation

Take the total energy in kilojoules provided, which is 1670 kJ, and divide it by the conversion factor 4.184 kJ/Cal. \[\text{Dietary Calories} = \frac{1670 \text{ kJ}}{4.184 \text{ kJ/Cal}} = 399.14 \text{ Cal}.\]
03

Round the Answer

Round the calculated dietary Calories to an appropriate number of significant figures. Since 1670 has three significant figures, round 399.14 to three significant figures as well.\[399.14 \approx 399 \text{ Cal}.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dietary Calories
Dietary Calories, often symbolized as Cal (with an uppercase 'C'), are a measure of energy that we obtain from foods such as chocolate cake. When you look at a food label, the energy content is usually given in Calories. In scientific terms, 1 dietary Calorie is equivalent to 4.184 kilojoules (kJ). This conversion is vital when calculating how much energy a particular food item contains in different units.
You might also hear the term "kilocalories" being used interchangeably with Calories. That's because 1 Calorie is technically 1 kilocalorie (1000 calories with a lowercase 'c').
This energy conversion helps in balancing dietary intake with the body’s energy needs and understanding how much energy comes from various foods. Remember, knowing how to convert between units like kilojoules and Calories is key when evaluating nutritional content.
Significant Figures
Significant figures are a crucial concept in chemistry and physics that ensure precision in our calculations. They indicate the certainty or reliability of a measurement.
In the exercise, the energy provided by the chocolate cake was given as 1670 kJ. This figure has three significant numbers (1, 6, and 7).
When performing calculations, it is important to maintain the same level of precision. In this case, we calculated the dietary Calories as 399.14, but because the original kilojoules were presented with three significant figures, the final answer must also adhere to this precision, becoming 399 Cal.
  • This precision ensures that we don't imply more accuracy than our measurements allow.
  • Always round your final answer to match the number of significant figures of your initial data.
Kilojoules
Kilojoules (kJ) are a unit of energy commonly used outside of the United States to quantify the energy content of foods. In the metric system, joules are the standard unit for energy, and a kilojoule is simply a multiple of a thousand joules.
When converting other units of energy like Calories, kilojoules provide a more scientific metric. A single dietary Calorie converts to exactly 4.184 kJ, which can be helpful in various calculations involving energy.
In nutrition, understanding kilojoules helps in comparing the energy content of foods when labeled in different regions. Converting between kilojoules and Calories often requires dividing or multiplying by this conversion factor, making it essential for everyone calculating energy intake and expenditure in food science and dietetics.

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Most popular questions from this chapter

Adding \(5.44 \mathrm{g}\) of \(\mathrm{NH}_{4} \mathrm{NO}_{3}(\mathrm{s})\) to \(150.0 \mathrm{g}\) of water in a coffee-cup calorimeter (with stirring to dissolve the salt) resulted in a decrease in temperature from \(18.6^{\circ} \mathrm{C}\) to \(16.2^{\circ} \mathrm{C} .\) Calculate the enthalpy change for dissolving \(\mathrm{NH}_{4} \mathrm{NO}_{3}(\mathrm{s})\) in water, in \(\mathrm{kJ} / \mathrm{mol}\). Assume that the solution (whose mass is \(155.4 \mathrm{g}\) ) has a specific heat capacity of \(4.2 \mathrm{J} / \mathrm{g} \cdot \mathrm{K} .\) (Cold packs take advantage of the fact that dissolving ammonium nitrate in water is an endothermic process.)

A 0.692 -g sample of glucose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6},\) is burned in a constant volume calorimeter. The temperature rises from \(21.70^{\circ} \mathrm{C}\) to \(25.22^{\circ} \mathrm{C} .\) The calorimeter contains \(575 \mathrm{g}\) of water and the bomb has a heat capacity of \(650 \mathrm{J} / \mathrm{K}\). What quantity of heat is evolved per mole of glucose?

For each of the following, define a system and its surroundings and give the direction of heat transfer between system and surroundings. (a) Methane is burning in a gas furnace in your home. (b) Water drops, sitting on your skin after a dip in a swimming pool, evaporate. (c) Water, at \(25^{\circ} \mathrm{C},\) is placed in the freezing compartment of a refrigerator, where it cools and eventually solidifies. (d) Aluminum and \(\mathrm{Fe}_{2} \mathrm{O}_{3}(\mathrm{s})\) are mixed in a flask sitting on a laboratory bench. A reaction occurs, and a large quantity of heat is evolved.

Calculate the quantity of heat required to convert \(60.1 \mathrm{g}\) of \(\mathrm{H}_{2} \mathrm{O}(\mathrm{s})\) at \(0.0^{\circ} \mathrm{C}\) to \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) at \(100.0^{\circ} \mathrm{C} .\) The heat of fusion of ice at \(0^{\circ} \mathrm{C}\) is \(333 \mathrm{J} / \mathrm{g}\); the heat of vaporization of liquid water at \(100^{\circ} \mathrm{C}\) is \(2260 \mathrm{J} / \mathrm{g}\).

The standard enthalpy of formation of solid barium oxide, \(\mathrm{BaO},\) is \(-553.5 \mathrm{kJ} / \mathrm{mol},\) and the enthalpy of formation of barium peroxide, \(\mathrm{BaO}_{2},\) is \(-634.3 \mathrm{kJ} / \mathrm{mol}\). (a) Calculate the standard enthalpy change for the following reaction. Is the reaction exothermic or endothermic? $$\mathrm{BaO}_{2}(\mathrm{s}) \longrightarrow \mathrm{BaO}(\mathrm{s})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g})$$ (b) Draw an energy level diagram that shows the relationship between the enthalpy change of the decomposition of \(\mathrm{BaO}_{2}\) to \(\mathrm{BaO}\) and \(\mathrm{O}_{2}\) and the enthalpies of formation of \(\mathrm{BaO}(\mathrm{s})\) and \(\mathrm{BaO}_{2}(\mathrm{s})\).

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