Question

The following table gives the vapor pressure of mercury at various temperatures. Fill in the rest of the table, and use the data to plot curves of \(P_{\text {vap }}\) versus \(T\) and \(\ln P_{\text {vap }}\) versus \(1 / T\). $$ \begin{array}{lccc} \hline \text { Temp }(\mathrm{K}) & P_{\text {vap }}(\mathrm{mm} \mathrm{Hg}) & \ln P_{\text {vap }} & 1 / T \\ \hline 500 & 39.3 & ? & ? \\ 520 & 68.5 & ? & ? \\ 540 & 114.4 & ? & ? \\ 560 & 191.6 & ? & ? \\ 580 & 286.4 & ? & ? \\ 600 & 432.3 & ? & ? \end{array} $$

Short answer

Complete the table using natural logs and reciprocal temperatures, and plot the curves.

Step-by-step solution

Calculate ln P_vap

To find \( \ln P_{\text{vap}} \) for each data point, take the natural logarithm of the given vapor pressure \( P_{\text{vap}} \). Use the formula:\[ \ln P_{\text{vap}}(\text{mm Hg}) = \ln(\text{given value}) \] For example, at 500 K, \( \ln P_{\text{vap}} = \ln(39.3) \approx 3.6742 \). Perform this calculation for each temperature value.

Calculate 1/T

For each temperature \( T \), compute \( \frac{1}{T} \) using the formula:\[ \frac{1}{T}(\text{K}) = \frac{1}{\text{temperature in Kelvin}} \] For example, at 500 K, \( \frac{1}{T} = \frac{1}{500} = 0.0020 \). Repeat this calculation for all temperature values.

Fill in the Table

Using the calculations from Step 1 and Step 2, fill in the missing values in the table:\[\begin{array}{lccc} \hline \text { Temp }(\mathrm{K}) & P_{\text {vap }}(\mathrm{mm} \mathrm{Hg}) & \ln P_{\text {vap }} & 1 / T \ \hline 500 & 39.3 & 3.6742 & 0.0020 \ 520 & 68.5 & 4.2288 & 0.001923 \ 540 & 114.4 & 4.7403 & 0.001852 \ 560 & 191.6 & 5.2546 & 0.001786 \ 580 & 286.4 & 5.6569 & 0.001724 \ 600 & 432.3 & 6.0684 & 0.001667 \end{array} \]

Plot P_vap vs. T Curve

Create a graph with temperature \( T \) on the x-axis and vapor pressure \( P_{\text{vap}} \) on the y-axis. Plot the points using the temperature and \( P_{\text{vap}} \) values from the table, then connect the points with a smooth curve. This graph will show how vapor pressure varies with temperature.

Plot ln P_vap vs. 1/T Curve

Create a second graph with \( \frac{1}{T} \) on the x-axis and \( \ln P_{\text{vap}} \) on the y-axis. Plot the points using the calculated \( \ln P_{\text{vap}} \) and \( \frac{1}{T} \) values, then connect the points with a smooth curve to show the relation between these variables.

Key concepts

Natural Logarithm

The natural logarithm, commonly denoted as \( \ln \), is a mathematical function that determines the logarithm to the base \( e \), where \( e \) is approximately 2.71828. This function is crucial in many scientific equations, particularly those involving rates and growths, making it very applicable in chemistry and physics.
When dealing with vapor pressure, taking the natural logarithm of pressure values (\( P_{\text{vap}} \)) can help simplify the relationships, permitting easier calculations and graphing, especially when investigating exponential changes with temperature.

• In this exercise, the calculation involves taking \( \ln \) of each given vapor pressure.
• For example, if the vapor pressure is 39.3 mm Hg, then \( \ln(39.3) \approx 3.6742 \).

Each value of \( \ln P_{\text{vap}} \) effectively narrows down the data range and provides a less steep, more manageable curve for graph plotting. Understanding how logarithms work is essential as they convert multiplicative processes to additive ones, simplifying the way we handle data transformations.

Plotting Graphs

Graphs are invaluable tools in science, allowing us to visualize and interpret data trends efficiently. When plotting vapor pressure data, two types of graphs are particularly insightful: \( P_{\text{vap}} \) versus temperature \( T \), and \( \ln P_{\text{vap}} \) versus \( 1/T \).
To create a clear and precise graph:

• The x-axis typically represents our independent variable such as temperature \( T \) or its inverse \( 1/T \).
• While the y-axis displays the dependent variable such as vapor pressure \( P_{\text{vap}} \) or its natural logarithm \( \ln P_{\text{vap}} \).

When plotting a graph of vapor pressure versus temperature, it's often non-linear and steep, emphasizing rapid changes in pressure as temperature increases. By contrast, plotting \( \ln P_{\text{vap}} \) against \( 1/T \) typically results in a linear graph. This linear relationship helps in analyzing the properties like enthalpy of vaporization using methods such as the Clausius-Clapeyron equation.

Temperature Dependence

Temperature is a critical factor influencing a substance's vapor pressure. As temperature increases, kinetic energy among molecules also rises, resulting in more molecules escaping into a vapor state, which increases vapor pressure.
This exercise illustrates how significantly vapor pressure changes with temperature, highlighting the exponential nature of this relationship. This means:

• Small temperature changes can lead to large changes in vapor pressure.
• The relationship is not linear; instead, it follows an exponential curve.

To better analyze this relationship, scientists often use a modified version of the Van’t Hoff equation or the Clausius-Clapeyron equation. These methods involve the natural logarithm of vapor pressure plotted against the inverse of temperature. Such plots provide a clearer view of how temperature affects vapor pressure, leading to greater insights into molecular energetics and phase transitions.