Problem 49 - Chemistry The Central Science In Si Units

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?

Step-by-Step Solution

Answer

(a) Use \(C_m = c \times M\). (b) \(24.282 \text{ J/(mol}\cdot\text{K)}\). (c) You need the mass of the aluminum piece.

Step 1: Understand the Relationship

The specific heat, \(c\), is defined as the amount of heat per unit mass required to raise the temperature by one degree Celsius or Kelvin. The equation for specific heat is \(c = \frac{q}{m \Delta T}\), where \(q\) is heat absorbed, \(m\) is mass, and \(\Delta T\) is the temperature change.

Step 2: Define Molar Heat Capacity

Molar heat capacity, \(C_m\), is the heat required to raise the temperature of one mole of a substance by one degree Celsius or Kelvin. It is related to specific heat by taking into account the molar mass \(M\) (mass of one mole) of the substance. The equation is \(C_m = c \times M\).

Step 3: Derive the Conversion Equation

To convert the specific heat to molar heat capacity, multiply the specific heat \(c\) by the molar mass \(M\) of the substance: \[C_m = c \times M\]. This equation allows us to convert from specific heat (\(\text{J/(g}\cdot\text{K)}\)) to molar heat capacity (\(\text{J/(mol}\cdot\text{K)}\)).

Step 4: Calculate Aluminum's Molar Heat Capacity

The molar mass of aluminum is approximately \(26.98 \text{ g/mol}\). Using the conversion equation, \[C_m = 0.9 \text{ J/(g}\cdot\text{K)} \times 26.98 \text{ g/mol} = 24.282 \text{ J/(mol}\cdot\text{K)}\].

Step 5: Determine Additional Information Needed

To calculate the heat capacity of a specific piece of aluminum, you need the mass of that aluminum piece. Heat capacity \(C\) can be found using the formula \(C = c \times m\), where \(m\) is the mass of the aluminum piece.

Key Concepts

Specific Heat

The concept of Specific Heat is essential in understanding how different substances respond to thermal energy. Specific heat, denoted as \( c \), is the energy required to raise the temperature of a unit mass (usually in grams) of a substance by one degree Kelvin or Celsius. Imagine heating a pot of water versus a pot of oil; the water might take longer to heat up because it has a higher specific heat. Knowing the specific heat allows us to predict how much energy a material can absorb or release when its temperature changes.
It is expressed in units of \( ext{J/(g} \cdot \text{K)} \), which gives a clear picture of how much heat energy is needed for even a small change in temperature. The formula for specific heat is:

• \( c = \frac{q}{m \Delta T} \)

Where:

• \( q \) is the heat absorbed or released in Joules,
• \( m \) is the mass in grams,
• \( \Delta T \) is the temperature change in Kelvin or Celsius.

Understanding specific heat helps in various applications, from cooking to designing thermal systems in technology and engineering.

Heat Capacity

Heat Capacity is a broader concept that extends beyond just a specific unit mass like specific heat does. While specific heat measures the heat required for one gram of a substance, heat capacity, usually denoted \( C \), refers to the total heat energy needed to raise the temperature of an entire object or system by one degree Kelvin or Celsius.
The distinction is important: Heat capacity is dependent on the mass of the object or substance, whereas specific heat is an intrinsic property that doesn’t change with mass. The formula for calculating the heat capacity is:

• \( C = c \times m \)

Where:

• \( c \) is the specific heat,
• \( m \) is the mass of the substance or object.

Heat capacity is expressed in Joules per Kelvin (J/K). This parameter gives engineers and scientists crucial information on how much energy a material will absorb or release as a whole, making it vital in various fields such as thermodynamics, material science, and environmental studies.

Molar Mass

Molar Mass is a fundamental concept in chemistry that becomes particularly significant when converting specific heat into molar heat capacity. Molar mass, represented as \( M \), is the mass of one mole of a substance, usually in grams per mole (g/mol). It links the microscopic world of molecules to the macroscopic amounts used in the laboratory.
This conversion is important for understanding the energy needed to raise the temperature of one mole of a substance. The connection between specific heat and molar heat capacity \( C_m \) is straightforward:

• \( C_m = c \times M \)

Where:

• \( c \) is the specific heat,
• \( M \) is the molar mass of the substance.

When you know the specific heat of a substance and its molar mass, you can calculate the molar heat capacity, giving a deeper insight into the heat absorption for chemical reactions and processes. This concept is widely used in chemical reactions, metallurgy, and the design of industrial processes where precise thermal management is crucial.