Problem 478 - JEE Physics

Question

An ice-cream has a marked value of \(700 \mathrm{kcal}\). How many kilo-watt- hour of energy will it deliver to the body as it is digested \((\mathrm{J}=4.2 \mathrm{~J} / \mathrm{cal})\) (A) \(0.81 \mathrm{kwh}\) (B) \(0.90 \mathrm{kwh}\) (C) \(1.11 \mathrm{kwh}\) (D) \(0.71 \mathrm{kwh}\)

Step-by-Step Solution

Answer

The ice-cream delivers 0.82 kWh to the body when digested, and the closest option is (A) 0.81 kWh.

Step 1: Convert kilocalories to calories

To convert the energy in kilocalories to calories, multiply the given value by 1000: Energy = 700 kcal × 1000 cal/kcal = 700,000 cal

Step 2: Convert calories to Joules

Next, we need to convert this energy from calories to Joules using the provided conversion factor 1 cal = 4.2 J: Energy = 700,000 cal × 4.2 J/cal = 2,940,000 J

Step 3: Convert Joules to kilowatt-hours

Finally, we need to convert the energy from Joules to kilowatt-hours. We know that 1 kWh = 3,600,000 J. Therefore, Energy = \( \frac{2,940,000 \mathrm{J}}{3,600,000 \mathrm{J/kWh}} = 0.8167 \mathrm{kWh} \)

Step 4: Round the answer and choose the correct option

Round the result to two decimal places: Energy = 0.82 kWh Comparing our result to the available options, the closest value is (A) 0.81 kWh.

Key Concepts

Calorie to Joule Conversion

When dealing with thermodynamics or energy conversion exercises, understanding how to convert calories to joules is essential. The calorie is a unit of energy mainly used in food labeling, while the joule is the SI unit for energy.

• The relationship between calories and joules is established by the conversion factor: 1 calorie is equivalent to approximately 4.184 joules.
• In the exercise, this conversion was simplified using 1 calorie equaling 4.2 joules to ease calculations.

To convert kilocalories to joules, convert kilocalories to calories first by multiplying by 1000 (since 1 kilocalorie = 1000 calories), then multiply by the joule conversion factor. This method allows for consistency and accuracy when working within scientific domains ensuring that energy values move seamlessly between systems.

Kilowatt-hour Calculation

A kilowatt-hour (kWh) is a unit of energy representing one kilowatt (1000 watts) of power used for one hour. It's widely used in electric utility applications and is a crucial concept in understanding energy consumption.

• When converting joules to kilowatt-hours, you need the conversion factor where 1 kWh = 3,600,000 joules.
• This factor comes from the relationship that 1 watt equals 1 joule per second, and an hour has 3600 seconds.

To convert from joules to kilowatt-hours, divide the energy in joules by 3,600,000. This conversion allows for an understanding of how energy measurements translate into practical, everyday use, like what you see on an electricity bill.

Joule Conversion Factor

Conversion factors are bridges between different units and are vital for ensuring accuracy and understanding in energy calculations. The joule conversion factor specifically helps convert calories, commonly used in food energy, to joules, the scientific standard of energy measurement.

• Using the correct conversion factor is crucial. For the exercise, the factor 4.2 joules per calorie was used, signifying an approximation for ease.
• For more precision in experimental or high-accuracy conditions, use the exact conversion of 4.184 joules per calorie.

Employing correct conversion factors ensures reliable results and aids in conveying the energy value across different scientific and practical applications.

Rounding Off Calculations

Rounding is used to make numbers more manageable and to align with practical levels of precision. In many exercises, especially involving multiple conversions, rounding simplifies results, but must be done aptly to avoid errors or misrepresentations.

• In the exercise, the energy value was rounded to two decimal places to match the answer choices available, which is a common practice in problem-solving for clarity.
• Always follow the established norms or guidelines for rounding depending on the context, such as economic rounding or scientific rounding conventions.

Rounding helps in focusing on significant figures, reducing calculation-induced errors, and making data presentation cleaner and easier to interpret for real-world applications.