Suppose the vapor pressure of a substance is measured at two...

Suppose the vapor pressure of a substance is measured at two different temperatures. (a) By using the ClausiusClapeyron equation, Equation $11.1$, derive the following relationship between the vapor pressures, $P_{1}$ and $P_{2}$, and the absolute temperatures at which they were measured, $T_{1}$ and $T_{2}$ $$ \ln \frac{P_{1}}{P_{2}}=-\frac{\Delta H_{\mathrm{vap}}}{R}\left(\frac{1}{T_{1}}-\frac{1}{T_{2}}\right) $$ (b) Gasoline is a mixture of hydrocarbons, a major component of which is octane, $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2}$ $\mathrm{CH}_{2} \mathrm{CH}_{3} .$ Octane has a vapor pressure of $13.95$ torr at $25{ }^{\circ} \mathrm{C}$ and a vapor pressure of $144.78$ torr at $75^{\circ} \mathrm{C}$. Use these data and the equation in part (a) to calculate the heat of vaporization of octane. (c) By using the equation in part (a) and the data given in part (b), calculate the normal boiling point of octane. Compare your answer to the one you obtained from Exercise $11.86 .$ (d) Calculate the vapor pressure of octane at $-30^{\circ} \mathrm{C}$.

Suppose the vapor pressure of a substance is measured at two different temperatures. (a) By using the ClausiusClapeyron equation, Equation $11.1$, derive the following relationship between the vapor pressures, $P_{1}$ and $P_{2}$, and the absolute temperatures at which they were measured, $T_{1}$ and $T_{2}$
$$
\ln \frac{P_{1}}{P_{2}}=-\frac{\Delta H_{\mathrm{vap}}}{R}\left(\frac{1}{T_{1}}-\frac{1}{T_{2}}\right)
$$
(b) Gasoline is a mixture of hydrocarbons, a major component of which is octane, $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{2}$
$\mathrm{CH}_{2} \mathrm{CH}_{3} .$ Octane has a vapor pressure of $13.95$ torr at $25{ }^{\circ} \mathrm{C}$ and a vapor pressure of $144.78$ torr at $75^{\circ} \mathrm{C}$. Use these data and the equation in part (a) to calculate the heat of vaporization of octane. (c) By using the equation in part (a) and the data given in part (b), calculate the normal boiling point of octane. Compare your answer to the one you obtained from Exercise $11.86 .$ (d) Calculate the vapor pressure of octane at $-30^{\circ} \mathrm{C}$.

Key Concepts

Clausius-Clapeyron Equation

This thermodynamic relationship quantitatively links the change in vapor pressure with temperature for a phase transition. By integrating a differential form that relates the derivative of the logarithm of the pressure to the enthalpy of phase change, one can derive an expression that predicts how vapor pressure changes with inverse temperature. It is foundational for understanding the vaporization process, as it provides a means to calculate the heat of vaporization from experimental pressure–temperature data.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its condensed phases (liquid or solid) at a given temperature. It is a key concept in phase transitions because it determines conditions for boiling and condensation. Variations in vapor pressure with temperature, as described by the Clausius-Clapeyron equation, allow one to predict phase behavior under different thermal conditions.

Heat of Vaporization

The heat of vaporization is the amount of energy required to convert a substance from the liquid phase to its gaseous phase at constant temperature. It reflects the strength of intermolecular forces in the liquid; higher values indicate stronger attractions. In the context of the Clausius-Clapeyron equation, the heat of vaporization directly influences the slope in the pressure versus temperature relationship, thereby allowing its determination through experimental vapor pressures measured at different temperatures.

Normal Boiling Point

The normal boiling point is defined as the temperature at which the vapor pressure of a liquid equals the standard atmospheric pressure (1 atm). This concept is crucial in practical applications because it indicates the temperature at which a substance transitions from the liquid to the gas phase under standard conditions. Using the Clausius-Clapeyron relation, one can calculate the normal boiling point by determining the temperature at which the evaluated vapor pressure meets this atmospheric criterion.

Transcript

00:01 This is a pretty giant equation, and we're going to be talking about some derivations using the closcius clapperon equation.

00:10 We're given the following information.

00:13 Suppose the vapor pressure of a substance is measured at two different temperatures.

00:22 Part a says using the closcius clapperon equation, derive the equation i'm going to write down, relating vapor pressures and the absolute temperature at which they were.

01:19 Measure.

02:05 Okay, i'm not sure i'm going to be able to fit it on this page.

02:08 So i think i might just go to my next page and start writing.

02:15 Okay.

02:18 So the problem i'm given is, okay, the clausias -coparian equation is a little bit bigger.

06:38 Okay, so here is the equation we're trying to get, and here's the drive -to equation.

06:49 So we did part a.

06:56 I got to go to the next page in my text to get to part b.

07:13 Part b, gasoline is a mixture of hydrocarbons, and octane is a major component of that mixture.

07:49 I don't know why they're writing octane like this.

07:53 I'll write it like the text.

08:06 1, 2, 3, 4, 6, 7, 8.

08:15 1, 2, 3, 4, 5, 6, 7, 8.

08:18 Okay.

08:18 Okay, octane has a vapor pressure of 13 .95 tor, which is the same thing as millimeters of mercury, which you might be more familiar with, at 25 degrees celsius, and it has a vapor pressure of 144 .78 tor at 75 degrees celsius.

09:03 We use these data in the equation in part a to calculate the heat of vaporization.

09:26 Okay, b.

09:31 So we're going to, i wonder if i should write down that equation again.

09:36 See if i have room.

09:37 I think i do.

09:38 I'm going to write down my equation pretty small here, and i'll write that down in red.

10:08 Okay, so let's go ahead and punch our values or plug our values into this.

10:18 P1.

10:22 I'm going to leave my units off because those will cancel out.

10:37 This heat of vaporization is what i'm looking for.

11:11 These are kelvin temperatures.

11:57 Make my minus a little bigger.

12:54 Let me see here.

12:55 Make sure i've got this right.

13:13 Then to two sig figs, where am i going to go here? i think i'll go to three sig figs on here, 4 .04 times 10 to the fourth joules per mole.

13:41 And then if i want to report that in kilojoules, it'll be 40 .4 kilojoules per mole.

13:54 And that would be my heat of vaporization.

14:01 For octane.

14:04 Okay...

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