Question

Vapor pressure measurements at several different temperatures are shown here for mercury. Determine graphically the molar heat of vaporization for mercury. \(\begin{array}{cccccc}t\left({ }^{\circ} \mathrm{C}\right) & 200 & 250 & 300 & 320 & 340 \\ P(\mathrm{mmHg}) & 17.3 & 74.4 & 246.8 & 376.3 & 557.9\end{array}\)

Short answer

The molar heat of vaporization for mercury can be determined graphically using the Clausius-Clapeyron equation and linear regression. The exact value, however, depends on the computed slope from the regression line fitted to the data points of ln(P) versus 1/T graph.

Step-by-step solution

Plot the data

Make a plot using the given data with temperature (in Kelvin) on the x-axis and the natural logarithm of pressure on the y-axis. To convert temperature from Celsius to Kelvin, add 273.15 to each temperature value.

Apply linear regression

Apply linear regression to the plot in order to find the best-fit line. The properties of this line will be used to find the molar heat of vaporization. The best-fit line should minimize the sum of the squares of the vertical distances of the points from the line.

Determine the slope

From the best-fit line, compute the slope. Remember, in the equation ln(P) = -ΔHvap/(R*T) + C, the slope of the linear regression line (when plotted as ln(P) vs 1/T) is -ΔHvap/R.

Compute the molar heat of vaporization

Substitute the computed slope and the value of R into the equation ΔHvap = -R * slope to obtain the molar heat of vaporization. As a reminder, the value of the gas constant R is 8.314 J/(mol*K).

Key concepts

Vapor Pressure

Vapor pressure is an essential concept in physical chemistry, particularly when discussing phase transitions like vaporization. It refers to the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. The vapor pressure of a substance increases with temperature, as more molecules have enough kinetic energy to escape from the liquid or solid phase into the vapor phase.

In practice, understanding vapor pressure is crucial because it influences processes such as boiling and evaporation. When the vapor pressure of a liquid reaches the ambient pressure, the liquid will boil. Measurement of the vapor pressure across different temperatures can provide valuable information about the substance, such as its molar heat of vaporization, which is a measure of the energy required to vaporize one mole of the substance.

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation is a powerful tool in thermodynamics that describes the phase transition between two phases of matter, like the liquid and vapor phase. It is especially useful for estimating the vapor pressure of a liquid at different temperatures and for calculating the molar heat of vaporization.

This equation can be written as: \(\ln(P) = -\frac{\Delta H_{\text{vap}}}{R} \frac{1}{T} + C\) where \(\ln(P)\) is the natural logarithm of the vapor pressure, \(\Delta H_{\text{vap}}\) is the molar heat of vaporization, \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, and \(C\) is a constant related to the entropy of vaporization. By graphing the natural logarithm of vapor pressure against the reciprocal of the temperature, we can determine the molar heat of vaporization as the slope of the resulting line multiplied by the negative of the gas constant.

Linear Regression

Linear regression is a statistical method that models the relationship between two variables by fitting a linear equation to observed data. It is an indispensable technique in experimental chemistry for analyzing how one variable changes with another. In the context of determining the molar heat of vaporization, linear regression is used to find the best-fit line through a plot of the natural logarithm of vapor pressure against the reciprocal of the temperature.

Once the data is plotted, linear regression calculates the line that minimizes the sum of the squares of the distances (residuals) of the data points from the line. The slope of this line, when graphed as \(\ln(P)\) versus \(1/T\), relates directly to the molar heat of vaporization through the Clausius-Clapeyron equation. Therefore, accurately applying linear regression to the data is crucial to obtaining a reliable measure of the molar heat of vaporization.

Natural Logarithm

The natural logarithm, often abbreviated as \(\ln\), is the logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. In many scientific calculations, the natural logarithm is used because of its natural properties when dealing with growth processes and exponential decay, as well as its occurrence in many natural phenomena.

In the context of the Clausius-Clapeyron equation, the natural logarithm allows us to linearize the relationship between vapor pressure and temperature, enabling the use of linear regression to determine the molar heat of vaporization. Understanding how to work with natural logarithms is essential for students tackling exercises involving the Clausius-Clapeyron equation and thermodynamic calculations, as it can simplify complex relationships into a form that is much easier to analyze graphically and mathematically.