Question

Use the following data to estimate \(\Delta H\) for the reaction \(\mathrm{S}^{-}(g)+\) \(\mathrm{e}^{-} \rightarrow \mathrm{S}^{2-}(g)\). Include an estimate of uncertainty. $$ \begin{array}{|lcccc|} \hline & & \text { Lattice } & & \Delta H_{\text {sub }} \\ & \Delta \boldsymbol{H}_{\mathrm{t}}^{\circ} & \text { Energy } & \text { I.E. of } \mathbf{M} & \text { of M } \\ \hline \mathrm{Na}_{2} \mathrm{~S} & -365 & -2203 & 495 & 109 \\ \mathrm{~K}_{2} \mathrm{~S} & -381 & -2052 & 419 & 90 \\ \mathrm{Rb}_{2} \mathrm{~S} & -361 & -1949 & 409 & 82 \\ \mathrm{Cs}_{2} \mathrm{~S} & -360 & -1850 & 382 & 78 \\ \hline \end{array} $$ $$ \begin{aligned} \mathrm{S}(s) & \longrightarrow \mathrm{S}(g) & \Delta H &=227 \mathrm{~kJ} / \mathrm{mol} \\ \mathrm{S}(g)+\mathrm{e}^{-} & \longrightarrow \mathrm{S}^{-}(g) & \Delta H &=-200 \mathrm{~kJ} / \mathrm{mol} \end{aligned} $$ Assume that all values are known to \(\pm 1 \mathrm{~kJ} / \mathrm{mol}\).

Short answer

Using the Born-Haber cycle and linear regression relationships from the provided data, we estimate the unknown \(\Delta H\) for the reaction \(\mathrm{S}^{-}(g) + \mathrm{e}^{-} \rightarrow \mathrm{S}^{2-}(g)\) to be around \(\Delta H_t^\circ\) kJ/mol, with an uncertainty of \(\pm 1\) kJ/mol.

Step-by-step solution

Identify Relationships

For estimating the unknown \(\Delta H\), we can make use of linear regression relationships from the given table data to find a relationship between lattice energy, ionization energy, and sublimation energy. The relationships are as follows: 1. Lattice energy (\(\Delta H_t^\circ\)) and sublimation energy (\(\Delta H_\text{sub}\)). 2. Lattice energy (\(\Delta H_t^\circ\)) and ionization energy (I.E. of M).

Perform Linear Regression

Carry out linear regression for both the relationships identified in Step 1. For example, create a scatter plot or use any statistical software to find the best-fit line represented by: 1. Lattice energy vs. sublimation energy: \(\Delta H_t^\circ = a_1 (\Delta H_\text{sub}) + b_1\) 2. Lattice energy vs. ionization energy: \(\Delta H_t^\circ = a_2 (\text{I.E. of M}) + b_2\) Obtain the slope and intercept for both the relationships (\(a_1, b_1, a_2,\) and \(b_2\)).

Estimate Unknown \(\Delta H\)

Using the relationships obtained in Step 2, we can now estimate the unknown \(\Delta H\) for the following reaction: \(\mathrm{S}^{-}(g) + \mathrm{e}^{-} \rightarrow \mathrm{S}^{2-}(g)\) The lattice energy for this reaction is \(\Delta H_t^\circ\), and it will be related to both the sublimation energy (\(\Delta H_\text{sub}\)) and the ionization energy (I.E.). By plugging the values for the reaction \(\mathrm{S}(g) + \mathrm{e}^{-} \rightarrow \mathrm{S}^{-}(g)\), which has \(\Delta H_\text{sub} = 227\) kJ/mol and I.E. of \(-200\) kJ/mol, into the best-fit line equations obtained from Step 2: 1. \(\Delta H_t^\circ = a_1 (227) + b_1\) 2. \(\Delta H_t^\circ = a_2 (-200) + b_2\) Calculate the value of \(\Delta H_t^\circ\) in both cases.

Calculate Average and Uncertainty

Now, find the average value of \(\Delta H_t^\circ\) for the unknown reaction: $$\Delta H_t^\circ = \frac{\Delta H_t^\circ(1) + \Delta H_t^\circ(2)}{2}$$ For uncertainty, we know that all values are known to \(\pm 1\) kJ/mol. Hence, plug these uncertainties into the relationships from Step 2 to calculate the error range for \(\Delta H_t^\circ\). The final uncertainty value for \(\Delta H_t^\circ\) should be the maximum error from the obtained error ranges.

Report Results

Now that we have estimated the unknown \(\Delta H\) and the associated uncertainty, we can report the result as: $$ \Delta H = \Delta H_t^\circ \pm \text{uncertainty} \, \mathrm{(kJ/mol)} $$ This completes the estimation of \(\Delta H\) for the given reaction, along with its associated uncertainty.

Key concepts

Lattice Energy

Lattice energy is a term referring to the energy released when oppositely charged ions in the gaseous state come together to form a crystalline solid. It is a measure of the strength of the bonds in that ionic compound. The larger the lattice energy, the stronger the forces holding the ions together, and as a result, the more stable the compound is. In an educational setting, we often talk about lattice energy when dealing with the formation of ionic compounds, which is a critical concept in thermodynamics and chemical bonding.

For estimating enthalpy changes in reactions involving ionic compounds, such as the formation of an ion from a neutral atom, it's necessary to consider the lattice energy as part of the total energy calculation. It's important to know that the lattice energy is not directly measurable; instead, it's calculated using theoretical models or derived from experimental data through the Born-Haber cycle.

Ionization Energy

Ionization energy is defined as the amount of energy required to remove an electron from a neutral atom or molecule in the gaseous state to form a cation. It is a reflection of the attraction between the nucleus and the electrons: the higher the ionization energy, the more difficult it is to remove an electron from the atom. This concept is foundational to understanding chemical reactivity and trends in the periodic table, where elements are organized partly by their ionization energy.

As it's linked to the electron configuration and the effective nuclear charge, ionization energies generally increase across a period in the periodic table as the nuclear charge and hold on the electrons strengthen. Ionization energy is crucial for predicting the behavior of atoms during chemical reactions, especially in processes like oxidation and reduction.

Sublimation Energy

Sublimation energy is the energy required to change a substance from its solid phase directly into the gas phase without passing through the liquid phase, at a given pressure (usually atmospheric pressure). It is a characteristic thermodynamic property of substances that under specific conditions, tend to skip the liquid state, such as dry ice (solid carbon dioxide).

The sublimation process is endothermic, meaning it absorbs heat from its surroundings. This energy goes into breaking the intermolecular forces that hold the solid together. An understanding of sublimation energy is important for various chemical processes, including purification (like that of iodine), deposition techniques in materials science, and the explanation of natural phenomena like the sublimation of ice in comets.

Linear Regression Analysis

Linear regression analysis is a statistical tool that models the relationship between two variables by fitting a linear equation to the observed data. It is widely used in scientific research, including chemistry, to understand how different variables correlate with each other. For instance, a linear regression can help predict unknown values within a system by establishing a mathematical relationship between known quantities.

In the context of estimating enthalpy changes, linear regression can be used to derive relationships between properties like lattice energy and sublimation energy, or ionization energy, which can then be used to predict unknown enthalpies. The simplicity and effectiveness of linear regression analysis make it a valuable method for interpreting experimental data and driving insights in scientific inquiry.

Enthalpy of Sublimation

Enthalpy of sublimation is the total amount of heat energy absorbed when a substance transitions directly from the solid phase to the gas phase at a constant pressure, usually considered to be atmospheric pressure. This thermodynamic quantity is related to the sublimation energy but includes the pressure-volume work done against the atmosphere during the phase change.

Because it involves heating a solid to the point where its molecules have enough energy to overcome their intermolecular forces and disperse as a gas, the enthalpy of sublimation is a critical consideration for materials that skip the liquid phase under standard conditions. When dealing with thermal energetic calculations such as the one in the exercise provided, the enthalpy of sublimation is considered when calculating the overall enthalpy change for a reaction involving the sublimation step.

Uncertainty Calculation

In scientific measurements and calculations, uncertainty quantifies the range of values within which the true value is expected to lie. Whenever we make a measurement or calculation, we must acknowledge the precision of our instruments and the limitations in our methods, which are expressed in terms of uncertainty. This is a critical component of data analysis, as it allows us to understand the reliability of our results.

In estimating enthalpy changes, as was required in the given exercise, incorporating the uncertainties for each of the measured quantities allows for a more accurate statement of the result. When calculating uncertainties, we usually take into account the precision of the measuring device, any errors in measurement, and the reproducibility of the results. In the case of enthalpy calculation using linear regression, the uncertainty in the prediction must also encompass the uncertainty in the slope and the intercept of the regression line, thus providing a range within which the true enthalpy change would reasonably fall.