Problem 107 - Chemical Principles

Question

The vapor pressure of phosphoryl chloride difluoride \(\left(\mathrm{OPClF}_{2}\right)\) has been measured as a function of temperature: \begin{tabular}{cc} Temperature (K) & Vapor pressure (Torr) \\ \hline \(190 .\) & \(3.2\) \\ 228 & 68 \\ \(250 .\) & \(240 .\) \\ 273 & 672 \\ \hline \end{tabular} (a) Plot \(\ln P\) against \(T^{-1}\) (this project is best done with the aid of a computer or a graphing calculator that can calculate a leastsquares fit to the data). (b) From the plot (or a linear equation derived from it) in part (a), determine the standard enthalpy of vaporization of \(\mathrm{OPClF}_{2} ;(c)\) the standard entropy of vaporization of \(\mathrm{OPClF}_{2}\); and (d) the normal boiling point of \(\mathrm{OPClF}_{2}\). (c) If the pressure of a sample of \(\mathrm{OPClF}_{2}\) is reduced to 15 Torr, at what temperature will the sample boil?

Step-by-Step Solution

Answer

Perform a linear regression of ln P vs. 1/T to find the slope and y-intercept, which will give the standard enthalpy and entropy of vaporization, respectively. Use the regression equation to find the normal boiling point by solving for T when P is 760 Torr and the boiling temperature at 15 Torr by solving for T when P is 15 Torr.

Step 1: Plot Ln P versus 1/T

Collect the data points from the table and transform the vapor pressures (P) into their natural logarithms (ln P). Then, find the reciprocal of each temperature in Kelvin (1/T). Use a computer or graphing calculator to plot ln P vs. 1/T and perform a least-squares fit to obtain the best straight-line approximation to the data.

Step 2: Determine the Standard Enthalpy of Vaporization

The slope of the line from the ln P vs. 1/T plot is equal to \( -\Delta H_{vap} / R \), where \( R \) is the universal gas constant (8.314 J/mol*K). Calculate \( -\Delta H_{vap} \) by multiplying the slope by \( -R \).

Step 3: Calculate the Standard Entropy of Vaporization

The y-intercept of the straight-line fit is equal to \( \Delta S_{vap} / R \). Multiply the y-intercept by \( R \) to obtain the standard entropy of vaporization \( \Delta S_{vap} \).

Step 4: Determine the Normal Boiling Point

The normal boiling point is the temperature at which the vapor pressure equals 1 atm (760 Torr). Solve the equation obtained from the linear fit (ln P = -\Delta H_{vap}/R \cdot 1/T + \Delta S_{vap}/R) for \( T \) when \( P \) is 760 Torr. Convert ln(760) into \( 1/T_{b} \) using the linear equation.

Step 5: Calculate the Boiling Temperature at 15 Torr

To find the temperature at which the sample will boil when the pressure is reduced to 15 Torr, insert ln(15) into the linear plot and solve for \( T \) to find the new boiling temperature.

Key Concepts

Standard Enthalpy of Vaporization

The standard enthalpy of vaporization, often denoted as \( \Delta H_{vap} \), is a measure of the energy required to convert one mole of a substance from the liquid phase to the vapor phase at a constant temperature, typically at the substance's boiling point under standard pressure (1 atm or 101.325 kPa). This enthalpy change is usually expressed in kilojoules per mole (kJ/mol).

Understanding \( \Delta H_{vap} \) is crucial for students because it provides insights into the strength of the intermolecular forces within the liquid. A higher value of \( \Delta H_{vap} \) suggests stronger forces that need more energy to overcome. When students learn how to plot \( \ln P \) against \( T^{-1} \) and calculate the slope, they're essentially determining \( -\Delta H_{vap} / R \) where \( R \) is the gas constant. By multiplying the negative slope by \( -R \), they find the actual \( \Delta H_{vap} \) value, unraveling the energy dynamics of vaporization.

Standard Entropy of Vaporization

The standard entropy of vaporization, \( \Delta S_{vap} \), represents the change in entropy when one mole of a liquid vaporizes at a fixed temperature and pressure (standard conditions). The entropy of a substance is a measure of its disorder or randomness; thus, \( \Delta S_{vap} \) illustrates how much more disordered the molecules become as they transition from a liquid to a gaseous state.

This concept is essential as it complements the enthalpy of vaporization in predicting the spontaneity of the phase change. By analyzing the y-intercept of the plot of \( \ln P \) versus \( T^{-1} \) as taught, which is \( \Delta S_{vap} / R \), students can multiply this intercept by the gas constant \( R \) to find the actual \( \Delta S_{vap} \). This calculation aids in understanding energy distribution and molecular randomness during vaporization.

Normal Boiling Point

The normal boiling point of a substance is the temperature at which its vapor pressure is equal to the standard atmospheric pressure of 1 atm. It's a critical parameter linked to both the standard enthalpy and entropy of vaporization. When students are performing calculations to find the normal boiling point, they're identifying a specific temperature where a liquid's transition to vapor under normal atmospheric conditions occurs.

To determine this temperature, students set the vapor pressure \( P \) to 1 atm in their Clausius-Clapeyron derived linear equation and solve for temperature \( T \). Practically, this translates to using the linear relationship established from the plot of \( \ln P \) against \( T^{-1} \) to find the reciprocal of the temperature \( 1/T_b \) corresponding to \( \ln(760 \) Torr) as Torr is another unit of pressure where 760 Torr equals 1 atm.

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation is a foundational principle in thermodynamics that describes how the vapor pressure of a substance varies with temperature. This relation underpins the concept that vapor pressure increases with temperature, reflecting increased molecular motion and tendency to escape the liquid phase.

In an educational setting, teaching the Clausius-Clapeyron equation reinforces students' understanding of phase transitions and the relation between temperature, vapor pressure, and entropy. By rearranging the equation \( \ln P = -\Delta H_{vap}/R \cdot 1/T + \Delta S_{vap}/R \) and plotting \( \ln P \) versus \( 1/T \) to obtain a straight line, students witness firsthand how the equation models vapor pressure behavior across different temperatures. Grasping this concept, they can predict the boiling temperature at various pressures, a practical skill in fields such as chemical engineering and materials science.