Show that the Clausius-Clapeyron equation can be written as lnP=(-ΔHvap)/(R T)+B where B is a co

step 1 In this question, first of all, we need to define what the slope of this line is. So there's a t

Show that the Clausius-Clapeyron equation can be written as...

Show that the Clausius-Clapeyron equation can be written as $$\ln P=\frac{-\Delta H_{\mathrm{vap}}}{R T}+B$$ where $B$ is a constant that has different values for different substances. This is an equation for a straight line. (a) What is the expression for the slope of this line? (b) Using the following vapor pressure data, plot $\ln P$ vs. $1 / T$ for ethyl acetate, $\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}$, a common organic solvent used in nail polish removers. $\begin{array}{lccccc}t\left({ }^{\circ} \mathrm{C}\right) & -43.4 & -23.5 & -13.5 & -3.0 & +9.1 \\ \mathrm{vp} \text { (torr) } & 1 & 5 & 10 . & 20 . & 40 . \\ t\left({ }^{\circ} \mathrm{C}\right) & +16.6 & +27.0 & +42.0 & +59.3 & \end{array}$ $\begin{array}{lllll}\text { (torr) } & 60 . & 100 . & 200 . & 400\end{array}$ (c) From the plot, estimate $\Delta H_{\mathrm{vap}}$ for ethyl acetate. (d) From the plot, estimate the normal boiling point of ethyl acetate.

Show that the Clausius-Clapeyron equation can be written as
$$\ln P=\frac{-\Delta H_{\mathrm{vap}}}{R T}+B$$
where $B$ is a constant that has different values for different substances. This is an equation for a straight line. (a) What is the expression for the slope of this line? (b) Using the following vapor pressure data, plot $\ln P$ vs. $1 / T$ for ethyl acetate, $\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}$, a common organic solvent used in nail polish removers. $\begin{array}{lccccc}t\left({ }^{\circ} \mathrm{C}\right) & -43.4 & -23.5 & -13.5 & -3.0 & +9.1 \\ \mathrm{vp} \text { (torr) } & 1 & 5 & 10 . & 20 . & 40 . \\ t\left({ }^{\circ} \mathrm{C}\right) & +16.6 & +27.0 & +42.0 & +59.3 & \end{array}$
$\begin{array}{lllll}\text { (torr) } & 60 . & 100 . & 200 . & 400\end{array}$
(c) From the plot, estimate $\Delta H_{\mathrm{vap}}$ for ethyl acetate.
(d) From the plot, estimate the normal boiling point of ethyl acetate.

Key Concepts

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation is a fundamental thermodynamic relation that describes the phase boundary between two phases of matter (typically liquid and vapor). It relates the rate of change of vapor pressure with temperature to the enthalpy of phase transition, providing insight into how pressure and temperature are interdependent during processes like evaporation or boiling.

Enthalpy of Vaporization

Enthalpy of vaporization is the heat required to convert a unit quantity of a substance from its liquid phase into its vapor phase at constant pressure. This quantity appears in the Clausius-Clapeyron equation and determines the steepness of the pressure-temperature relationship, as a higher enthalpy leads to a greater sensitivity of vapor pressure to changes in temperature.

Linearization of Thermodynamic Equations

The linearization of the Clausius-Clapeyron equation involves taking the natural logarithm of the vapor pressure and plotting it against the reciprocal of temperature. This transformation converts an exponential relationship into a linear one, where the slope and intercept of the line can be used to extract meaningful thermodynamic parameters, including the enthalpy of vaporization and boiling point information.

Graphical Analysis in Thermodynamics

Graphical analysis, particularly plotting ln(P) versus 1/T, is a valuable technique in thermodynamics. It allows experimental data to be linearized so that parameters such as the slope (related to the enthalpy of vaporization) and intercept can be determined accurately, thus facilitating the estimation of key thermodynamic properties from empirical measurements.

Phase Equilibria and Boiling Point

Phase equilibria deals with the balance between different phases of a substance, such as liquid and vapor. The boiling point is defined as the temperature at which the vapor pressure of a liquid equals the external pressure. By analyzing the linear plot obtained from the Clausius-Clapeyron equation, one can extrapolate to determine the boiling point and understand the conditions under which phase transitions occur.

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