Question

Under special conditions, sulfur reacts with anhydrous liquid ammonia to form a binary compound of sulfur and nitrogen. The compound is found to consist of $69.6\mathrm{\%}\mathrm{S}$ and $30.4\mathrm{\%}$ N. Measurements of its molecular mass yield a value of $184.3\mathrm{g}{\mathrm{mol}}^{-1}.$ The compound occasionally detonates on being struck or when heated rapidly. The sulfur and nitrogen atoms of the molecule are joined in a ring. All the bonds in the ring are of the same length. (a) Calculate the empirical and molecular formulas for the substance. (b) Write Lewis structures for the molecule, based on the information you are given. (Hint: You should find a relatively small number of dominant Lewis structures.) (c) Predict the bond distances between the atoms in the ring. (Note: The $\mathrm{S}-\mathrm{S}$ distance in the $S_8$ ring is $2.05\text{AA}$.) (d) The enthalpy of formation of the compound is estimated to be $480\mathrm{kJ}{\mathrm{mol}}^{-1}$. $\mathrm{\Delta }{H}_f^{\circ }$ of $S(g)$ is $222.8\mathrm{kJ}{\mathrm{mol}}^{-1}$. Estimate the average bond enthalpy in the compound.

Short answer

The molecular formula of the compound is $S_4{N}_4$, and its most stable Lewis structure consists of alternating double bonds between sulfur and nitrogen atoms, satisfying the octet rule with no lone pairs. The bond distances for $\mathrm{S}-\mathrm{N}$ and $\mathrm{S}-\mathrm{S}$ are both close to $2.05\text{AA}$. The average bond enthalpy in the compound is around -51.4 kJ/mol.

Step-by-step solution

Calculate Empirical Formula

To find the empirical formula of the compound, first, calculate the moles of sulfur and nitrogen present in the 100g sample of the compound, since the percentage composition of S and N is given. Moles of S \(= \frac{69.6 \mathrm{~g}}{\mathrm{32.07 \mathrm{~g} \mathrm{~mol}^{-1}} = 2.17\) moles Moles of N \( = \frac{30.4 \mathrm{~g}}{\mathrm{14.01 \mathrm{~g} \mathrm{~mol}^{-1}} = 2.17\) moles Since the moles of Sulfur and Nitrogen are equal(2.17), the empirical formula is $SN$.

Calculate Molecular Formula

Given the molecular weight of the compound is $184.3\mathrm{g}{\mathrm{mol}}^{-1}$. Calculate the weight of the empirical formula (SN): $32.07\mathrm{g}/\mathrm{mol}+14.01\mathrm{g}/\mathrm{mol}=46.08\mathrm{g}/\mathrm{mol}$ Now, to find how many times the empirical formula weight fits into the molecular weight: $\frac{184.3}{46.08}\approx 4$ So, the molecular formula is: $S_4{N}_4$

Construct Lewis Structures

From the molecular formula $S_4{N}_4$, we can see that there are a total of 8 atoms in the ring structure. Each nitrogen atom contributes 5 valence electrons, and each sulfur atom contributes 6 valence electrons, totaling $(5\times 4)+(6\times 4)=44$ valence electrons. By forming single bonds between each sulfur and nitrogen atom for all, we will use 8 electrons, leaving 36 electrons. Distributing the remaining electrons to fulfill the octet rule, the most stable Lewis structure for S and N alternating in the ring would involve the formation of double bonds between adjacent S and N atoms. This satisfies the octet rule for each atom and leaves no lone pairs on any of the atoms.

Predict Bond Distances

Since all the bonds in the ring are double bonds, the bond lengths between atoms are expected to be in the range of double bond distances. The given $\mathrm{S}-\mathrm{S}$ distance in the $S_8$ ring is $2.05\text{AA}$. In this case, we can assume that the bond distance $\mathrm{S}-\mathrm{N}$ should be similar, as the molecule is symmetric and consists of alternating double bonds in the ring, hence, the bond distances for both $\mathrm{S}-\mathrm{N}$ and $\mathrm{S}-\mathrm{S}$ bonds are close to $2.05\text{AA}$.

Calculate the average bond enthalpy

The enthalpy of formation of the compound is provided: $\mathrm{\Delta }{H}_f^{\circ }({\mathrm{S}}_4{\mathrm{N}}_4)=480\mathrm{kJ}{\mathrm{mol}}^{-1}$. The enthalpy of formation of S is given as $\mathrm{\Delta }{H}_f^{\circ }(\mathrm{S}(\mathrm{g}))=222.8\mathrm{kJ}{\mathrm{mol}}^{-1}$. The enthalpy of formation of N is $\mathrm{\Delta }{H}_f^{\circ }(\mathrm{N}(=\mathrm{N}))/2=\frac{N_2(g)}{2}=0$ since it is an element in its standard state. Total enthalpies of formation for 4 atoms of sulfur and 4 molecules of nitrogen are: $4\times 222.8\mathrm{kJ}+4\times 0\mathrm{kJ}=891.2\mathrm{kJ}$. The change in the enthalpy of formation, $\mathrm{\Delta }{H}_{rxn}$, is the difference between the formation enthalpies of products and reactants: $\mathrm{\Delta }{H}_{rxn}=\mathrm{\Delta }{H}_f^{\circ }({\mathrm{S}}_4{\mathrm{N}}_4)-(4\times \mathrm{\Delta }{H}_f^{\circ }(\mathrm{S}(\mathrm{g})))=480\mathrm{kJ}{\mathrm{mol}}^{-1}-891.2\mathrm{kJ}{\mathrm{mol}}^{-1}=-411.2\mathrm{kJ}{\mathrm{mol}}^{-1}$ To find the average bond enthalpy, divide $\mathrm{\Delta }{H}_{rxn}$ by the number of bonds in the ring (8): $\frac{-411.2\mathrm{kJ}{\mathrm{mol}}^{-1}}{8}=\text{-51.4 kJ/mol }$ The average bond enthalpy in the compound is around -51.4 kJ/mol.

Key concepts

Empirical Formula Calculation

The empirical formula is the simplest whole-number ratio of elements in a compound. It is derived based on the percentage composition of elements. In this exercise, the compound consists of sulfur and nitrogen where sulfur makes up 69.6% and nitrogen 30.4%.
To find the empirical formula, we need to calculate the number of moles of each element in a 100g sample. For sulfur, the molar mass is 32.07 g/mol, and for nitrogen, it's 14.01 g/mol.

• Moles of sulfur: $\frac{69.6\text{ g}}{32.07\text{ g/mol}}=2.17\text{ moles}$
• Moles of nitrogen: $\frac{30.4\text{ g}}{14.01\text{ g/mol}}=2.17\text{ moles}$

This results in a 1:1 ratio of sulfur to nitrogen. Therefore, the empirical formula is $\text{SN}$.
Empirical formulas are important because they provide the basic ratio of the elements in a compound, which can help in determining other characteristics, such as the molecular formula.

Molecular Structure

Molecular structure describes the arrangement of atoms within a molecule. It goes beyond the simple formula by detailing how the atoms are bonded. For this compound, the molecular formula obtained is ${\text{S}}_4{\text{N}}_4$, indicating there are 4 sulfur and 4 nitrogen atoms.
The structure is a ring, with all the atoms connected in a circular pattern as stated in the problem. Here, each bond in the molecule is the same length, suggesting a stable and symmetrical molecular structure.
The actual shape and orientation of a molecule impact its properties and behavior, including reactivity and interaction with other molecules.

Bond Enthalpy

Bond enthalpy, or bond dissociation energy, refers to the amount of energy required to break a bond between two atoms in a molecule. It is typically measured in kilojoules per mole (kJ/mol) and is an indicator of bond strength.
In this case, the average bond enthalpy was calculated based on the provided enthalpies of formation. Using the difference in enthalpy changes, the compound shows an average bond enthalpy of $-51.4\text{ kJ/mol}$. This negative value indicates that the formation of the compound from gaseous sulfur and nitrogen releases energy, highlighting the compound's stability under normal conditions.
Bond enthalpies can give insight into the robustness of a compound and its energy changes during reactions.

Chemical Bonding

Chemical bonding involves the interactions that hold atoms together in molecules. For ${\text{S}}_4{\text{N}}_4$, these include covalent bonds, likely involving both $\text{S}-\text{S}$ and $\text{S}-\text{N}$ bonds in the ring structure.
In covalent bonding, atoms share pairs of electrons to achieve stable electron configurations mimicking noble gases. The bond lengths being equal suggests the involvement of either single or double bonds regularly repeating around the ring.
Chemical bonding is key to understanding molecular shape, compound stability, and how molecules interact with each other.

Lewis Structures

Lewis structures are diagrams that represent the valence electrons of atoms within a molecule, illustrating how atoms are bonded and any lone pairs present. For ${\text{S}}_4{\text{N}}_4$, constructing a Lewis structure requires arranging sulfur and nitrogen atoms in a ring.
Each sulfur atom has six valence electrons, and each nitrogen atom has five. Using these, we form bonds by sharing electrons. Initially, form single bonds between all atoms, using 8 electrons, and then add more bonds to satisfy the octet rule. As stated, the structure achieves stability by forming alternating double bonds between the sulfur and nitrogen atoms. This arrangement satisfies the electron requirements for all atoms without needing extra lone pairs.
Understanding Lewis structures helps to predict molecular geometry and reactivity, as they depict the initial step in recognizing the compound's complete electronic structure.