Perform each conversion. (a) \(25 \mathrm{kWh}\) to joules (b) 249 cal to Calories (c) \(113 \mathrm{cal}\) to kilowatt-hours (d) \(44 \mathrm{~kJ}\) to calories

The conversion from kilowatt-hours (kWh) to joules (J) is a fundamental concept in energy conversion chemistry. A kilowatt-hour is a unit of energy that represents the amount of work done when a power of one kilowatt is applied for one hour. To convert from kWh to joules, we use the standard conversion factor where one kilowatt-hour is equivalent to 3.6 million joules, expressed as 1 kWh = 3.6 x 10^6 J.

For instance, to convert 25 kWh to joules, you would perform the following calculation: \[ 25 \text{kWh} \times 3.6 \times 10^6 \text{J/kWh} = 90 \times 10^6 \text{J} \] This step-by-step process allows for an exact conversion from kWh to joules, ensuring that energy values are correctly translated between the two units.

Understanding the difference between 'cal' and 'Calories' is crucial when dealing with dietary energy content. In chemistry and physics, a calorie (with a lowercase 'c') is defined as the energy required to raise the temperature of one gram of water by one degree Celsius. However, in nutritional contexts, we use the term 'Calories' with a capital 'C', also referred to as 'kilocalories' (\text{kcal}), where 1 Calorie = 1000 calories.

To convert cal to Calories, simply divide the amount in calories by 1000. For example, converting 249 cal to Calories involves the following: \[ 249 \text{cal} \div 1000 = 0.249 \text{Calories} \] This conversion is a key element in understanding nutritional labels and accurately tracking energy intake.

Another important conversion in understanding energy usage and efficiency is from calories to kilowatt-hours (kWh). To perform this conversion, we need two steps: converting calories to joules and then joules to kilowatt-hours. As we remember, 1 cal is equivalent to 4.184 J. Hence, to convert calories to kWh, the calories are first converted to joules, and then joules to kWh using the factor where 1 kWh equals 3.6 x 10^6 joules.

For an example of 113 cal to kWh: \[ 113 \text{cal} \times 4.184 \text{J/cal} = 472.792 \text{J} \] Then, to convert joules to kWh: \[ 472.792 \text{J} \div (3.6 \times 10^6 \text{J/kWh}) = 0.0001313322222 \text{kWh} \] This result can be rounded to an appropriate number of significant figures, depending on the precision required for the task at hand.

The process of converting kilojoules to calories is another important task, especially when comparing different units of energy. A kilojoule (kJ) is a unit of energy in the International System of Units (SI) and is equal to 1000 joules. Since 1 cal equals 4.184 joules, this conversion requires two steps: converting from kJ to J, and then from J to cal.

For converting 44 kJ to calories, the calculation is as follows: \[ 44 \text{kJ} \times 1000 \text{J/kJ} \div 4.184 \text{J/cal} = 10502.3923444976 \text{cal} \] This conversion is very useful in fields like biology and nutrition, where energy content in food is frequently expressed in calories. This understanding allows for more accurate calculations of energy consumption and dietary needs.