Question

Given the following thermochemical equations, $$ 3 \mathrm{Mg}(s)+2 \mathrm{NH}_{3}(g) \longrightarrow \mathrm{Mg}_{3} \mathrm{~N}_{2}(s)+3 \mathrm{H}_{2}(g) $$ \(\Delta H^{\circ}=-371 \mathrm{~kJ}\) $$ \frac{1}{2} \mathrm{~N}_{2}(g)+\frac{3}{2} \mathrm{H}_{2}(g) \longrightarrow \mathrm{NH}_{3}(g) \quad \Delta H^{\circ}=-46 \mathrm{~kJ} $$ calculate \(\Delta H^{\circ}\) (in kilojoules) for the following reaction: $$ 3 \mathrm{Mg}(s)+\mathrm{N}_{2}(g) \longrightarrow \mathrm{Mg}_{3} \mathrm{~N}_{2}(s) $$

Short answer

\(\Delta H^{\circ} = 93.5 \mathrm{kJ}\)

Step-by-step solution

Identify Relevant Equations

We have two given thermochemical equations:(1) \(3\mathrm{Mg}(s)+2\mathrm{NH}_3(g) \rightarrow \mathrm{Mg}_3\mathrm{N}_2(s)+3\mathrm{H}_2(g)\) with \(\Delta H^{\circ}=-371\mathrm{~kJ}\)(2) \(\frac{1}{2}\mathrm{N}_2(g)+\frac{3}{2}\mathrm{H}_2(g) \rightarrow \mathrm{NH}_3(g)\) with \(\Delta H^{\circ}=-46\mathrm{~kJ}\)We are tasked with finding the \(\Delta H^{\circ}\) for the target reaction:(3) \(3\mathrm{Mg}(s)+\mathrm{N}_2(g) \rightarrow \mathrm{Mg}_3\mathrm{N}_2(s)\)

Determine Adjustments to Equations

To find the \(\Delta H^{\circ}\) for reaction (3), we have to adjust the given equations (1) and (2) to sum up to the target equation. This involves reversing and scaling the equations as needed.For Equation (2), we need two moles of \(\mathrm{NH}_3\) for Equation (1), so we must multiply Equation (2) by 4, giving us:\(2\mathrm{N}_2(g) + 6\mathrm{H}_2(g) \rightarrow 4\mathrm{NH}_3(g)\) with \(\Delta H^{\circ}=-46 \mathrm{kJ} \times 4 = -184 \mathrm{kJ}\). Next, reverse Equation (1) because we need \(\mathrm{N}_2\) and \(\mathrm{H}_2\) as reactants to cancel out with the products of Equation (2).

Apply Hess's Law

Sum the modified Equations (1) and (2) to obtain Equation (3). Reverse Equation (1) changes the sign of \(\Delta H^{\circ}\):\(-1\times[3\mathrm{Mg}(s)+2\mathrm{NH}_3(g) \rightarrow \mathrm{Mg}_3\mathrm{N}_2(s)+3\mathrm{H}_2(g)]\) giving \(\Delta H^{\circ}=+371 \mathrm{kJ}\).Addition of the equations gives us:\(2\mathrm{N}_2(g) + 6\mathrm{H}_2(g) \rightarrow 4\mathrm{NH}_3(g)$$+\mathrm{Mg}_3\mathrm{N}_2(s)+3\mathrm{H}_2(g) \rightarrow 3\mathrm{Mg}(s)+2\mathrm{NH}_3(g)$$2\mathrm{N}_2(g) + 3\mathrm{H}_2(g) \rightarrow 3\mathrm{Mg}(s) + \mathrm{Mg}_3\mathrm{N}_2(s)\) which is twice the desired result, so we divide the entire reaction by 2.

Calculate the Total Enthalpy Change

Finally, calculate the total enthalpy change by adding the adjusted \(\Delta H^{\circ}\) values for the two equations:\(\Delta H^{\circ}\)(target reaction) = \(\frac{1}{2} \times (371 \mathrm{kJ} - 184 \mathrm{kJ})$$\Delta H^{\circ}\)(target reaction) = \(\frac{1}{2} \times 187 \mathrm{kJ}$$\Delta H^{\circ}\)(target reaction) = \(93.5 \mathrm{kJ}\)

Key concepts

Hess's Law

Hess's Law is a fundamental concept in chemical thermodynamics stating that the total enthalpy change in a chemical reaction is the same, no matter how the reaction takes place, as long as initial and final conditions are identical. It's based on the principle that enthalpy, symbolized as \(H\), is a state function, which means its value is determined only by the current state of the system, not the path used to reach that state.

When approaching a complex reaction that cannot be easily measured experimentally, Hess's Law allows us to use known enthalpy changes of related reactions to determine the enthalpy change of the target reaction. This comes in handy in educational contexts or when working in a laboratory to figure out the energetics of reactions where direct measurement is impractical. In essence, by breaking down a complex reaction into a series of simpler steps for which enthalpy changes are known, we can sum up the enthalpies of these steps according to the law to find the overall enthalpy change.

Enthalpy Change Calculation

The calculation of enthalpy change, denoted as \( \Delta H\), represents the heat absorbed or released under constant pressure during a chemical reaction. This quantity is central to energy considerations in chemical reactions, indicating whether a process is endothermic (heat absorbed, \( \Delta H > 0\)) or exothermic (heat released, \( \Delta H < 0\)).

To calculate the enthalpy change for a given reaction, you must often manipulate and combine known thermochemical equations, adjusting coefficients to reflect the molar ratios of the reactants and products as they appear in the reaction of interest. It's essential to remember that when a reaction is reversed, the sign of \( \Delta H\) also reverses, and when a reaction is multiplied by a coefficient, the \( \Delta H\) value should be multiplied by the same factor. These adjustments align with Hess's Law, allowing us to rearrange and sum the \( \Delta H\) values for each step to obtain the final enthalpy change.

Chemical Thermodynamics

Chemical thermodynamics is the branch of chemistry that deals with the relationships between heat, work, temperature, and chemical reactions or physical changes of state. It encompasses the principles that govern the direction and speed at which chemical reactions occur. Important concepts within this field include the laws of thermodynamics, enthalpy (\(H\)), entropy (\(S\)), and free energy (\(G\)).

When studying chemical reactions, understanding thermodynamics allows scientists to predict how energy transfers from one system to another, the spontaneity of reactions, and the equilibrium position that a reaction may achieve. For students tackling exercises in chemical thermodynamics, it's crucial to grasp these principles to explain why certain reactions occur and what factors can influence them, including temperature, pressure, and the presence of catalysts. With a robust foundation in these concepts, students can more readily comprehend the energy aspects of chemistry that drive so many processes we observe in both the laboratory and the natural world.