Problem 83 - Chemistry

Question

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

Step-by-Step Solution

Answer

The molar heat of vaporization for mercury can be found by converting the temperatures to Kelvin, plotting the natural log of the pressures against the inverses of the temperatures, performing a linear regression to find the best-fit line's slope, and then using this slope to find the molar heat of vaporization using \(\Delta H_{vap} = -m \cdot R\).

Step 1: Change of Units

First, the given temperatures need to be converted from Celsius to Kelvin. This can be done using the equation \(T(K) = t(°C) + 273.15\).

Step 2: Transformation of Data

Plot the natural logarithm of the pressure values, \(\ln(P)\), as a function of the inverse of the temperature in Kelvin, \(\frac{1}{T}\). The Clausius-Clapeyron equation prescribes these transformations: \( \ln(P) = -\frac{\Delta H_{vap}}{R}\left(\frac{1}{T}\right) + C \), where \(\Delta H_{vap}\) is the molar heat of vaporization, \(R\) is the ideal gas constant, and \(C\) is a constant. This equation form indicates that the plot of \(\ln(P)\) against \(\frac{1}{T}\) should be linear.

Step 3: Linear Regression

Perform a linear regression of the data. This means finding the best-fit line to the points in the plot from the previous step. In the linear equation \(y = mx + c\), \(m\) represents the slope of this line.

Step 4: Deduction of Molar Heat

The negative slope of the line in the plot, \(-m\), equals \(\frac{\Delta H_{vap}}{R}\). Therefore, multiply the slope by -R to get the molar heat of vaporization, \(\Delta H_{vap}\).

Key Concepts

Vapor Pressure

Vapor pressure is a fundamental concept in thermodynamics and is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature in a closed system. It represents how volatile a substance is; the higher the vapor pressure, the more readily the substance vaporizes. For a substance like mercury, vapor pressure increases with temperature, a pattern observable in the data provided.

When measuring vapor pressure experimentally, several temperatures are chosen to understand how vapor pressure changes. In the exercise, vapor pressure values of mercury at different temperatures are given. The relationship between temperature and vapor pressure is not linear but can be linearized using the Clausius-Clapeyron equation, which then allows the molar heat of vaporization to be determined graphically. Understanding vapor pressure is critical for a variety of applications, such as distillation, where it helps in determining the boiling point of a liquid.

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation provides a way to quantify vapor pressure at different temperatures and to estimate thermodynamic quantities like the molar heat of vaporization. It is an exponential relationship described by the equation \( \[ \ln(P) = -\frac{\Delta H_{vap}}{R}\left(\frac{1}{T}\right) + C \] \), where \(P\) is vapor pressure, \(\Delta H_{vap}\) is the molar heat of vaporization, \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, and \(C\) is a constant that represents the integration constant from the equation's derivation.

The equation tells us that the natural log of the vapor pressure is linearly related to the inverse of the temperature, a relationship that can be exploited to determine \(\Delta H_{vap}\) graphically by plotting \(\ln(P)\) against \(\frac{1}{T}\). The slope of this line is related to \(\Delta H_{vap}\), making it possible to calculate the molar heat of vaporization once the slope is found.

Linear Regression

Linear regression is a statistical tool used to create a straight line (best-fit line) through a set of points on a graph, to show a relationship between two variables. In the context of determining the molar heat of vaporization, a plot of the natural logarithm of vapor pressure (\ln(P)) against the inverse of temperature (\frac{1}{T}) should yield a straight line if the Clausius-Clapeyron equation holds true.

The best-fit line equation typically takes the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. The process of linear regression calculates the values of \(m\) and \(c\) that minimize the distance between the data points and the line, providing the most statistically accurate depiction of the trend. By performing linear regression on temperature and vapor pressure data, the slope of the line gives us the necessary information to calculate the molar heat of vaporization, as it is directly proportional to the slope.

Temperature Conversion

Temperature conversion is a crucial step in most thermodynamic calculations, as the temperature scales commonly used in different contexts may differ. For example, Celsius is a common scale for everyday situations, but Kelvin is the standard unit of temperature in scientific and thermodynamic equations, like the Clausius-Clapeyron equation.

To convert from Celsius to Kelvin, the conversion formula \(T(K) = t(°C) + 273.15\) is used. This allows us to input the correct temperature value into thermodynamic equations. Misinterpreting or overlooking temperature conversion can lead to significant calculation errors. In the context of our exercise, ensuring accurate temperature conversion allows for the proper plotting and analysis of vapor pressure against temperature, which is necessary to find the molar heat of vaporization for mercury.