Problem 95 - Chemistry

Question

A 20 fluid oz. soda contains 238 Calories. (a) How many kilojoules does the soda contain? (b) For how many hours could the amount of energy in the soda light a 75 watt lightbulb? \((1 \mathrm{watt}=1 \mathrm{~J} / \mathrm{s})\)

Step-by-Step Solution

Answer

(a) The soda contains 996.992 kJ. (b) It could light a 75-watt bulb for about 3.692 hours.

Step 1: Convert Calories to Kilocalories

Since 1 Calorie (Cal) is equivalent to 1 kilocalorie (kcal), which equals 238 kcal in this case, we use this conversion to proceed with the calculation.

Step 2: Convert Kilocalories to Kilojoules

Use the conversion factor: 1 kcal = 4.184 kJ. Multiply the kilocalories by this factor: \[238 \text{ kcal} \times 4.184 \text{ kJ/kcal} = 996.992 \text{ kJ}\]

Step 3: Calculate Energy in Joules

Convert kilojoules to joules for the subsequent calculation:\[996.992 \text{ kJ} = 996,992 \text{ J}\] since 1 kJ = 1,000 J.

Step 4: Calculate Time to Power a Lightbulb

Determine how long the 75-watt bulb can be powered with this energy. Since power (P) in watts is energy (E) consumed per unit time (t) in seconds, use:\[t = \frac{E}{P} = \frac{996,992 \text{ J}}{75 \text{ W}}\]Calculate:\[t = 13,292.8 \text{ seconds}\]

Step 5: Convert Seconds to Hours

Convert total seconds to hours by dividing by 3600 (the number of seconds in an hour):\[13,292.8 \text{ seconds} \div 3600 \text{ seconds/hour} = 3.692 \text{ hours}\]

Key Concepts

Energy Conversion

Energy conversion is a fundamental concept that involves changing one form of energy into another. Understanding this concept is crucial in various fields and applications such as engineering, physics, and environmental science. Let's look at a practical example: converting calories (a unit of energy) into joules, which is a more universally recognized unit.

• Caloric Energy: In the nutritional sense, when we talk about calories, we're often dealing with kilocalories (kcal). One kilocalorie is equal to 1,000 calories we'll consume in food.
• Joule Energy: The joule (J) is the SI unit of energy. It is commonly used in physics to quantify the amount of work done or energy transferred.
• Conversion Factors: For our exercise, 1 kilocalorie equals 4.184 kilojoules (kJ). This conversion factor is crucial for converting between these units accurately.

These conversions are used commonly in energy calculations, and understanding them can help when looking at energy outputs of foods or fuels. Whenever you're working with energy, remember the conversion factor to seamlessly transition between different units.

Kilocalories to Kilojoules

Conversion from kilocalories to kilojoules is an essential part of energy calculations, especially in nutrition and energy consumption studies. If you’re given a value in kilocalories, here's how you convert it to kilojoules.

• Determine the Given: For example, a soda containing 238 kilocalories needs to be expressed in kilojoules.
• Use the Conversion Factor: The conversion factor from kilocalories to kilojoules is 4.184. This means 1 kcal = 4.184 kJ.
• Calculate: Multiply the number of kilocalories by the conversion factor: \[238 ext{ kcal} imes 4.184 ext{ kJ/kcal} = 996.992 ext{ kJ}\]This allows you to express energy content in the widely used joule-based system.

This conversion is vital not only in everyday dietary considerations but also in broader applications like engineering and physics, where energy measurements need to be precise.

Power Consumption Calculation

Understanding how to calculate power consumption is crucial for assessing how long an energy source can sustain an appliance or device. In the context of the soda energy, we can calculate how long the energy can light a bulb.First, recall that power can be expressed as energy divided by time. If you know the energy (in joules) and the power requirement (in watts), you can find the time duration by rearranging the formula:\[ t = \frac{E}{P} \]Here’s how the process works:

• Energy in Joules: Following previous conversions, 996,992 joules results from the energy content of the soda.
• Power Rating: A 75-watt bulb requires 75 joules per second to function.
• Calculate Duration: Insert values into the formula to find time: \[ t = \frac{996,992 ext{ J}}{75 ext{ W}} = 13,292.8 ext{ seconds} \]
• Convert to Hours: Since time is often more interpretable in hours, convert seconds to hours: \[ 13,292.8 ext{ seconds} \div 3600 ext{ seconds/hour} = 3.692 ext{ hours} \]

By understanding this calculation, you can assess how long your energy outputs can sustain various applications, a critical skill in effective energy management.