Question

(a) Derive an equation to convert the specific heat of a pure substance to its molar heat capacity. (b) The specific heat of aluminum is \(0.9 \mathrm{~J} /(\mathrm{g} \cdot \mathrm{K}) .\) Calculate its molar heat capacity. (c) If you know the specific heat of aluminum, what additional information do you need to calculate the heat capacity of a particular piece of an aluminum component?

Short answer

(a) The equation to convert specific heat (Cₚ) of a pure substance to its molar heat capacity (Cₚ,m) is: Cₚ,m = Cₚ × M, where M is the molar mass of the substance. (b) The molar heat capacity of aluminum is approximately 24.3 J/(mol·K), calculated using its specific heat (0.9 J/(g·K)) and molar mass (26.98 g/mol). (c) To calculate the heat capacity of a particular piece of an aluminum component, we need the mass of the aluminum piece in addition to the specific heat of aluminum.

Step-by-step solution

(a) Deriving the conversion equation

To convert specific heat (Cₚ; in J/(g·K) ) of a pure substance to its molar heat capacity (Cₚ,m; in J/(mol·K)), we can use the relationship: Cₚ,m = Cₚ × M where M is the molar mass of the substance (in g/mol).

(b) Calculating the molar heat capacity of aluminum

First, we need to know the molar mass of aluminum. From the periodic table, the molar mass of aluminum is 26.98 g/mol. Given the specific heat of aluminum as 0.9 J/(g·K), we can calculate the molar heat capacity of aluminum using the conversion equation derived in step (a): Cₚ,m = Cₚ × M Cₚ,m = \( 0.9 \frac{J}{g \cdot K} \times 26.98 \frac{g}{mol} \) Cₚ,m = \( 24.282 \frac{J}{mol \cdot K} \) Hence, the molar heat capacity of aluminum is approximately 24.3 J/(mol·K).

(c) Additional information required

To calculate the heat capacity of a particular piece of an aluminum component with a known specific heat, we also need to know the mass of the piece. The heat capacity can then be calculated as follows: Heat Capacity = mass × specific heat where mass is the mass of the aluminum piece (in grams) and specific heat is the specific heat of aluminum (in J/(g·K)).

Key concepts

Specific Heat

Specific heat is a measure of how much heat energy is required to raise the temperature of one gram of a substance by one degree Celsius (or Kelvin). It is a vital concept in thermodynamics and is expressed in units of Joules per gram per Kelvin (J/g·K). Understanding specific heat helps us understand how different materials respond to heat.

In practical terms, if a material has a high specific heat, it means the material can absorb a lot of heat without a significant change in temperature. Conversely, a low specific heat indicates the material heats up quickly. This property is crucial when considering materials for thermal management in engineering and construction applications.

Specific heat also plays a significant role in calculating other related properties, like molar heat capacity, which involves a different unit system focusing on moles instead of grams. Knowing the specific heat provides a foundational understanding necessary for further calculations in heat capacity.

Aluminum

Aluminum is a fascinating metal known for its low density, strength, and conductivity. It is widely used in industries such as aerospace, automotive, and consumer electronics due to its advantageous properties. One of the key thermal properties of aluminum is its specific heat, which is relatively high for a metal.

The specific heat of aluminum is given as 0.9 J/g·K, meaning it requires 0.9 Joules to raise the temperature of one gram of aluminum by one Kelvin. This property makes aluminum a suitable material for heat sinks and thermal management systems, where it absorbs and dissipates heat efficiently.

When working with aluminum in thermodynamic applications, its molar mass becomes important. Aluminum's molar mass is approximately 26.98 g/mol, which allows for conversion from specific heat to molar heat capacity, enabling more precise energy calculations involving chemical reactions or thermal management.

Heat Capacity Calculation

Heat capacity calculation is essential for determining how much heat energy is needed to change the temperature of a substance. The calculation involves using specific heat and the mass of the substance.

To compute heat capacity, you multiply the specific heat by the mass of the substance:

• Heat Capacity = mass × specific heat

This formula gives you the total heat capacity in Joules per Kelvin (J/K), which tells you the overall amount of heat the substance can absorb for each degree increase in temperature.

When calculating the heat capacity for a piece of aluminum, knowing the specific heat and the mass is crucial. For instance, if you have a 100 gram piece of aluminum, the heat capacity calculation would be:

Heat Capacity = 100 g × 0.9 J/g·K = 90 J/K

This calculation provides insight into thermal efficiency and energy needs, which is valuable for designing systems or components that interact with heat.