A. Norbornene, norbornadiene, nortricyclane, and...

a. Norbornene, norbornadiene, nortricyclane, and quadricyclane can all be hydrogenated to norbornane. The heats of hydrogenation are given in the chart. These data allow calculation of $\Delta H_{f}$ for the other derivatives and the results are given as Exp. in the table. The table also gives $\Delta H_{f}$ values calculated for each compound by $\mathrm{MM}$ and three semiemirical $\mathrm{MO}$ methods. Compare the accuracy of the semiempirical methods in predicting the experimental heats of formation. Subsequently, the same compounds were computed by ab initio and DFT methods. Isodesmic reactions were used to compare the $\Delta H_{f}$ of the compounds (except for G3(MP2), where atomization energies were used). Compare the ab initio and DFT results with the semiempirical results from Part (a). c. Heats of hydrogenation have also been calculated from the semiempirical data. Since the heats of hydrogenation include the $\Delta H_{f}$ of $\mathrm{H}_{2}$, which is zero, they can be calculated as follows: $$ \Delta H_{\mathrm{H} 2}=\Delta H_{f \text { product }}-\Delta H_{f \text { reactant }} $$ This leads to the calculated $\Delta H_{\mathrm{H} 2}$ shown below. The $\Delta H_{\mathrm{H} 2}$ can also be calculated on a strain compensation basis: $$ \Delta H_{\mathrm{H} 2}=\Delta H_{f \text { product }}-\Delta H_{f \text { reactant }}+\text { strain relief } $$ The calculated values are included in the table. Compare the calculated and experimental results.

a. Norbornene, norbornadiene, nortricyclane, and quadricyclane can all be hydrogenated to norbornane. The heats of hydrogenation are given in the chart. These data allow calculation of $\Delta H_{f}$ for the other derivatives and the results are given as Exp. in the table. The table also gives $\Delta H_{f}$ values calculated for each compound by $\mathrm{MM}$ and three semiemirical $\mathrm{MO}$ methods. Compare the accuracy of the semiempirical methods in predicting the experimental heats of formation.
Subsequently, the same compounds were computed by ab initio and DFT methods. Isodesmic reactions were used to compare the $\Delta H_{f}$ of the compounds (except for G3(MP2), where atomization energies were used). Compare the ab initio and DFT results with the semiempirical results from Part (a).
c. Heats of hydrogenation have also been calculated from the semiempirical data. Since the heats of hydrogenation include the $\Delta H_{f}$ of $\mathrm{H}_{2}$, which is zero, they can be calculated as follows:
$$
\Delta H_{\mathrm{H} 2}=\Delta H_{f \text { product }}-\Delta H_{f \text { reactant }}
$$
This leads to the calculated $\Delta H_{\mathrm{H} 2}$ shown below. The $\Delta H_{\mathrm{H} 2}$ can also be calculated on a strain compensation basis:
$$
\Delta H_{\mathrm{H} 2}=\Delta H_{f \text { product }}-\Delta H_{f \text { reactant }}+\text { strain relief }
$$
The calculated values are included in the table. Compare the calculated and experimental results.

Key Concepts

Isodesmic Reactions

Isodesmic reactions involve chemical reactions where the number and types of bonds remain constant from reactants to products. This approach is used in computational chemistry to minimize systematic errors when calculating relative energies, as many inherent errors cancel out, allowing for more reliable comparisons of heats of formation.

Density Functional Theory (DFT)

Density Functional Theory (DFT) is a computational quantum mechanical modeling method used to investigate the electronic structure of many-body systems, primarily in their ground state. DFT reinterprets the many-electron problem in terms of electron density, offering a cost-effective yet reasonably accurate alternative to ab initio methods for calculating molecular energies and properties.

Ab Initio Methods

Ab initio methods rely entirely on basic physical principles without using empirical parameters. These methods solve the Schrödinger equation directly for electrons in molecules, which typically offers higher accuracy in predicting thermodynamic properties such as heats of formation, though often at a greater computational cost.

Strain Compensation

Strain compensation accounts for the energy changes associated with the release or introduction of strain in molecules. In computational studies, incorporating strain compensation helps adjust calculated thermodynamic quantities, such as heats of hydrogenation, by explicitly considering the relief of ring strain or steric interactions that can significantly influence molecular energetics.

Heats of Hydrogenation

Heats of hydrogenation are the energy changes associated with the addition of hydrogen to unsaturated compounds. This measure is useful for assessing the degree of strain or stabilization present in cyclic or unsaturated systems, as the conversion to a saturated state often relieves strain and reflects differences in molecular stability.

Heats of Formation

Heats of formation represent the change in energy when a compound is formed from its constituent elements in their standard states. It is a fundamental thermodynamic parameter used to compare the intrinsic stability of different molecules, and it serves as a basis for calculating reaction energetics such as heats of hydrogenation.

Semiempirical Methods

Semiempirical methods are computational techniques that incorporate experimental data and empirical parameters into the quantum mechanical framework. They provide a balance between computational efficiency and accuracy, making them useful for estimating molecular properties like heats of formation, especially for larger systems where full ab initio methods may be too demanding computationally.

Transcript

0:00 Hi.

00:02 So we have two questions here.

00:03 First one, we are to use the data to calculate and plot the relative volatility of benzene with respect to cyclohexane versus benzene composition in the liquid phase.

00:15 And then we are to use the bond law equation to compute the activity coefficients over the entire range of composition and then compare them in aplot.

00:29 So the relative volatility of a substance is the measure of its tendency to vaporize.

00:35 And it is calculated as the ratio of the k value of the two substances.

00:42 So let's write that in here.

00:44 Relative volatility is equivalent to k value of second substance over k value of the first substance.

00:56 Now, k is the ratio of the mole fraction in vapor and liquid phases.

01:01 And you can write that in here.

01:05 K is equivalent to the mole fraction in.

01:09 Vapor phase y over the mole fraction in the liquid phase x so the relative volatility can be written as mole fraction in vapor phase over mole fraction in liquid phase this is for the second substance over y sub one over x sub one now for a two component mixture mole fractions add up to one so so that means y sub 1 is 1 minus y sub 2 and x sub 1 is 1 minus x sub 2.

01:53 And we can just rewrite this equation for the relative volatility as well in terms of y sub 2 and x sub 2.

02:07 So we have y sub 2 over x of 2 over 1 minus y sub 2 because y sub 1 is 1 minus y sub 2.

02:18 X sub 1 is 1 minus x sub 2.

02:22 So let's call this equation 1.

02:27 Now in this problem, we are provided with the mole fractions of benzene in both the liquid and vapor phases at various temperatures.

02:37 So we will use this equation to determine the relative volatility of, benzene with respect to cyclohexane and then we'll plot the results against the mole fraction of benzene in liquid.

02:52 So let me attach the resulting graph.

02:56 So again, we used equation one to find the values of the relative volatility.

03:06 So this is the column for the relative volatility.

03:12 And then we use this and and we plotted the results against the mole fraction of benzene.

03:23 This is a mole fraction of benzene in liquid.

03:29 So this is our answer for a.

03:31 This is the thought and the values of the relative volatility.

03:46 Now, there's a follow -up question.

03:48 What happens to the relative volatility in the vicinity of the astiotrope? so at the azeotrope, the mole fraction in the vapor and liquid phase is the same or very close to each other.

04:03 So that means y should be equivalent to x.

04:09 And from the table, we see it for benzene to be roughly at this volume.

04:20 So let's write that answer here at the azeotrope, the mole fraction in the vapor and liquid.

04:43 Face is equal and from the table we see it for benzene to be roughly at y sub 2 x2 .0 .545 and that of cyclohexane to be at y sub 1 .0 .455...

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