Problem 24 - Applied Calculus

Question

The specific heat, \(s,\) of an element is the number of calories of heat required to raise the temperature of one gram of the element by one degree Celsius. Use the following table to decide if \(s\) is proportional or inversely proportional to the atomic weight, \(w,\) of the element. If so, find the constant of proportionality.

Step-by-Step Solution

Answer

Check for constant \(s \times w\) to confirm inverse proportionality.

Step 1: Understand the definitions

Specific heat (\(s\)) of an element indicates how much energy is required to increase the temperature of one gram of the element by one degree Celsius. Proportionality between two variables, like \(s\) and the atomic weight \(w\), means either \(s\) is equal to a constant times \(w\) (direct proportionality) or equal to a constant divided by \(w\) (inverse proportionality).

Step 2: Organize and examine the data

List the specific heat and atomic weight of each element from the provided table. This will help in identifying any relationship between the two variables through their values.

Step 3: Test for direct proportionality

To test for direct proportionality, see if \( \frac{s}{w} \) for each element gives a constant value. If it is constant, then specific heat is directly proportional to atomic weight.

Step 4: Test for inverse proportionality

To test for inverse proportionality, calculate \(s imes w\) for each element. If the product is constant across the elements, then specific heat is inversely proportional to atomic weight.

Step 5: Analyze the results

If step 3 produced a constant value \( \frac{s}{w} \), specific heat is directly proportional to atomic weight. If step 4 produced a constant product \(s \times w\), then it is inversely proportional. The resulting constant from whichever holds true is the constant of proportionality.

Key Concepts

Proportionality in Specific Heat and Atomic Weight

When comparing two variables, it is important to understand the concept of proportionality. Proportionality refers to the relationship between two quantities where one is a constant multiple of the other. In the context of specific heat (\(s\)) and atomic weight (\(w\)) of an element, two types of proportional relationships can be identified:

• Direct Proportionality: This occurs when one variable is equal to a constant times the other. Mathematically, it is represented as \(s = k \cdot w\), where \(k\) is the constant of proportionality.
• Inverse Proportionality: This relationship exists when one variable is equal to a constant divided by the other, denoted as \(s = \frac{k}{w}\).

To determine the type of proportionality between specific heat and atomic weight, we need to assess these relationships by calculating the ratios (\(\frac{s}{w}\)) for direct proportionality or the products (\(s \cdot w\)) for inverse proportionality. If they yield a constant value across different elements, then a proportional relationship exists in the identified form.

Understanding Atomic Weight

Atomic weight plays a crucial role in determining the properties of an element, including its specific heat. Each element consists of atoms, and the atomic weight is essentially the average mass of the atoms in a chemical element, measured in atomic mass units (amu).

• Atomic weights are determined using isotopes and their abundance in nature.
• It tells us how heavy an atom of an element is, in comparison to other elements on the periodic table.
• Historical data reveals variations in atomic weight due to changes in isotopic abundance.

Understanding atomic weight is essential when analyzing how and why it affects specific heat. Elements with differing atomic weights will possess unique thermal properties. These differences can either play a role in determining the amount of energy needed to raise their temperature, influencing their specific heat values.

Exploring Calorimetry and Its Role in Measuring Specific Heat

Calorimetry is the science of measuring the amount of heat released or absorbed during a chemical reaction, phase transition, or heat capacity change. It is a key method in determining specific heat, particularly for substances where detailed insight is needed.

• Calorimeters: Special devices called calorimeters are used to measure the heat change in a system. These instruments are crucial for accurate calorimetry.
• Applications in Chemistry: By using calorimetry, chemists can delve into the intrinsic properties of elements, including specific heat capacity.
• Importance of Calibration: Proper calibration of a calorimeter ensures accurate measurements of specific heat.

In experiments, the amount of heat required to change the temperature of a material is directly measured, helping establish an understanding of specific heat. Through these measurements, scientists gain deeper insights into energy exchanges in different materials, often uncovering the influence of atomic structure and weight on thermal properties.