Question

Prepare a graph of specific heat capacities for metals versus their atomic weights. Combine the data in Figure 5.4 and the values in the following table. What is the relationship between specific heat capacity and atomic weight? Use this relationship to predict the specific heat capacity of platinum. The specific heat capacity for platinum is given in the literature as \(0.133 \mathrm{J} / \mathrm{g} \cdot \mathrm{K} .\) How good is the agreement between the predicted and actual values? $$\begin{aligned} &\begin{array}{|l|c|} \hline & \text { Specific Heat Capacity } \\ \text { Metal } & (\mathrm{J} / \mathrm{g} \cdot \mathrm{K}) \\ \hline \text { Chromium } & 0.450 \\ \text { Lead } & 0.127 \\ \text { Silver } & 0.236 \\ \text { Tin } & 0.227 \\ \text { Titanium } & 0.522 \end{array}\\\ &1 \end{aligned}$$

Short answer

The specific heat capacity tends to decrease with increasing atomic weight. The predicted value for platinum should be close to 0.133 J/g·K, with percentage difference indicating the accuracy.

Step-by-step solution

Gather Data

First, gather the data for atomic weights and specific heat capacities of the metals provided. Use the atomic weights from a reliable source like the periodic table: - Chromium: 51.996 g/mol - Lead: 207.2 g/mol - Silver: 107.868 g/mol - Tin: 118.71 g/mol - Titanium: 47.867 g/mol. Ensure these weights are associated correctly with their specific heat capacities.

Plot The Data

Using graphing software or graph paper, plot the specific heat capacities (y-axis) against the atomic weights (x-axis). Place a point on the graph for each metal using the collected data from Step 1. This will visually represent the relationship between specific heat capacity and atomic weight.

Analyze the Trend

Once all the data points are plotted, observe the pattern or trend formed. Typically, metals tend to show a trend where specific heat capacity decreases with increasing atomic weight. Look for a linear or other type of correlation to describe this relationship.

Determine the Relationship Equation

Using the best fit line or curve on the graph, determine the equation that represents the relationship between atomic weight and specific heat capacity. This may involve statistical tools or software to find the line of best fit equation, usually a simple linear regression.

Predict Platinum's Specific Heat

Using the equation derived from Step 4, substitute the atomic weight of platinum (approximately 195.084 g/mol) into the equation to predict its specific heat capacity.

Compare Prediction with Literature

Compare the predicted specific heat capacity value for platinum with the given literature value of 0.133 J/g·K. Assess the agreement by calculating the percentage difference: \[ \text{Percentage Difference} = \frac{|\text{Predicted} - \text{Actual}|}{\text{Actual}} \times 100 \% \] This will quantify how close the prediction is to the actual value.

Key concepts

Atomic Weight

Atomic weight, also known as atomic mass, is a fundamental property of elements. It represents the average mass of atoms of an element, measured in atomic mass units (amu), reflecting the natural isotopes' distribution. For each metal, the atomic weight can be found on the periodic table.
Understanding atomic weight is crucial because it affects many chemical properties, including specific heat capacity. A contrasting atomic weight in metals allows one to predict and compare various properties, such as thermal or electrical conductivity.
For metals in this exercise, their atomic weights are as follows:

• Chromium: 51.996 g/mol
• Lead: 207.2 g/mol
• Silver: 107.868 g/mol
• Tin: 118.71 g/mol
• Titanium: 47.867 g/mol

These values serve as key inputs in examining the relationship between atomic weight and specific heat capacity.

Metals

Metals are a class of elements characterized by good conductivity of heat and electricity, ductility, and malleability. In the context of specific heat capacity, metals exhibit certain patterns due to their atomic structure.
Specific heat capacity in metals is the amount of heat energy required to raise the temperature of one gram of metal by one degree Celsius. Each metal has a different specific heat, making this property useful for various applications, similar to alloys and thermal conductors.
In this exercise, the metals investigated are chromium, lead, silver, tin, and titanium. Each metal has a specific heat capacity, influencing how quickly they heat up or cool down.

• Chromium: 0.450 J/g·K
• Lead: 0.127 J/g·K
• Silver: 0.236 J/g·K
• Tin: 0.227 J/g·K
• Titanium: 0.522 J/g·K

Graphing Data

Graphing is a crucial step when examining relationships between variables. It visually represents data, providing insights into patterns or trends.
For this exercise, you plot specific heat capacities on the y-axis against atomic weights on the x-axis. This creates a clear picture of how these two variables interact.
This graph enables you to spot trends, such as whether the specific heat capacity decreases as atomic weight increases. Such visualization helps make informed predictions about unknown data points, like the specific heat capacity of platinum.
When graphing, ensure accuracy by carefully plotting each point based on the data collected in earlier steps. Using graphing software can enhance precision and support further analysis, like determining best-fit lines or correlation factors.

Linear Regression

Linear regression is a statistical tool used to model the relationship between a dependent variable and one or more independent variables. In this context, it helps establish the relationship between atomic weight and specific heat capacity.
Once data points from metals are plotted, linear regression can be used to find the best-fit line that minimizes discrepancies between the line and individual data points. The equation of this line provides a mathematical model that can predict the specific heat of unknown metals, like platinum.
Applying linear regression involves calculating the slope and intercept, which defines the line equation: \[ y = mx + b \]Here, \( y \) represents the specific heat capacity, \( m \) is the slope, \( x \) is the atomic weight, and \( b \) is the y-intercept.
This statistical technique quantifies the observed pattern and allows predictions based on correlations in your data. It involves a computational process often facilitated by statistical software.