Question

The specific heat of aluminum is \(0.22 \mathrm{cal} / \mathrm{g}-{ }^{\circ} \mathrm{C}\). What is the value of the specific heat in (a) \(\mathrm{kcal} / \mathrm{kg}-{ }^{\circ} \mathrm{C}\), (b) \(\mathrm{J} / \mathrm{kg}-{ }^{\circ} \mathrm{C}\) ? (Show your calculations.)

Short answer

(a) \(0.22 \mathrm{kcal} / \mathrm{kg}-^{\circ} \mathrm{C}\), (b) \(920.48 \mathrm{J} / \mathrm{kg}-^{\circ} \mathrm{C}\).

Step-by-step solution

Convert calories per gram to kilocalories per kilogram

We are given the specific heat in units of \(0.22 \mathrm{cal} / \mathrm{g}-^{\circ} \mathrm{C}\). To convert it to \(\mathrm{kcal} / \mathrm{kg}-^{\circ} \mathrm{C}\), note that 1 kilocalorie (kcal) is 1000 calories (cal), and 1 kilogram (kg) is 1000 grams (g).First, convert calories to kilocalories:\[0.22 \mathrm{cal} = 0.22 / 1000 \mathrm{kcal} = 0.00022 \mathrm{kcal}\]Next, convert gram to kilogram:Thus, the specific heat in \(\mathrm{kcal} / \mathrm{kg}-^{\circ} \mathrm{C}\) is:\[0.00022 \mathrm{kcal} / \mathrm{g}-^{\circ} \mathrm{C} \times 1000 \mathrm{g/kg} = 0.22 \mathrm{kcal} / \mathrm{kg}-^{\circ} \mathrm{C}\]

Convert kilocalories to joules

The specific heat in kilocalories per kilogram is \(0.22 \mathrm{kcal} / \mathrm{kg}-^{\circ} \mathrm{C}\). Knowing that 1 kilocalorie is equivalent to 4184 joules, we can convert kilocalories to joules:\[0.22 \mathrm{kcal} \times 4184 \mathrm{J/kcal} = 920.48 \mathrm{J}\]

Adjust for unit per kilogram

The conversion from \(0.22 \mathrm{kcal} / \mathrm{kg}-^{\circ} \mathrm{C}\) directly gives us the desired unit since we are converting each kilocalorie per kilogram, which is:\[0.22 \mathrm{kcal} / \mathrm{kg}-^{\circ} \mathrm{C} = 920.48 \mathrm{J} / \mathrm{kg}-^{\circ} \mathrm{C}\]

Key concepts

Aluminum Specific Heat

The specific heat of a material is a crucial property in thermodynamics, as it defines how much energy is needed to raise the temperature of a unit mass of that material by one degree Celsius. For aluminum, this property is relatively low at 0.22 cal/g-°C. This means aluminum heats up and cools down quickly compared to materials with a higher specific heat. When dealing with specific heat, it's important to remember:

• Specific heat capacity is different for each material.
• Aluminum’s specific heat is useful in applications like cooking, where rapid heating is necessary.
• Understanding these values helps in material selection for engineering and manufacturing processes.

Unit Conversion in Thermodynamics

In thermodynamics, unit conversion is a frequent task, particularly when dealing with energy, temperature, and mass. Converting between units allows you to apply physical constants and equations universally. For specific heat, conversions might include:

• Calories to Kilocalories: Understand that 1 kcal = 1000 cal. When converting a number like 0.22 cal, divide by 1000 to convert to kcal, resulting in 0.00022 kcal/g-°C.
• Grams to Kilograms: Similarly, 1 kg = 1000 g. Therefore, multiplying the specific heat in kcal/g-°C by 1000 g/kg converts it to a value appropriate for per kilogram units.
• Exact conversions ensure calculations remain consistent and yield correct results.

Calories to Joules Conversion

In many scientific calculations, we encounter the need to convert between calories and joules, as both are units of energy. Joules are part of the International System of Units (SI), while calories are commonly used in chemistry and biology. To convert calories to joules:

• Recognize that 1 cal is approximately 4.184 joules.
• Thus, for converting kilocalories to joules, use: 1 kcal = 4184 J.
• For the example given, at 0.22 kcal/kg-°C, the conversion to joules is: 0.22 kcal × 4184 J/kcal = 920.48 J/kg-°C.
• This knowledge ensures compatibility with other SI-based scientific calculations.