Given the following information: $$\frac{1}{2} \mathrm{N}_{2}(\mathrm{g})+\frac{3}{2} \mathrm{H}_{2}(\mathrm{g}) \longrightarrow \mathrm{NH}_{3}(\mathrm{g})\quad\quad\quad\quad\Delta H_{1}^{\circ}$$ $$\mathrm{NH}_{3}(\mathrm{g})+\frac{5}{4} \mathrm{O}_{2}(\mathrm{g}) \longrightarrow \mathrm{NO}(\mathrm{g})+\frac{3}{2} \mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \quad \Delta H_{2}^{\circ}$$ $$\mathrm{H}_{2}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \longrightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{l})\quad\quad\quad\Delta H_{3}^{\circ}$$ Determine \(\Delta H^{\circ}\) for the following reaction, expressed in terms of \(\Delta H_{1}^{\circ}, \Delta H_{2}^{\circ},\) and \(\Delta H_{3}^{\circ}\) $$\mathrm{N}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NO}(\mathrm{g}) \quad \Delta H^{\circ}=?$$

Step by step solution

01

Writing the desired equation

Identify the desired equation: \(\mathrm{N}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NO}(\mathrm{g})\)

02

Modifying the given equations

Transform the given equations into forms that will add in such a way to yield the desired equation. For the first equation, nothing needs to be done. For the second equation, it should be reversed and multiplied by \(2\) to produce \(-2 \mathrm{NH}_{3}(\mathrm{g}) - \frac{5}{2} \mathrm{O}_{2}(\mathrm{g}) \longrightarrow -2 \mathrm{NO}(\mathrm{g}) - 3 \mathrm{H}_{2}\mathrm{O}(\mathrm{l})\). The third equation should be multiplied by \(3\) to yield \(3 \mathrm{H}_{2}(\mathrm{g}) + \frac{3}{2} \mathrm{O}_{2}(\mathrm{g}) \longrightarrow 3 \mathrm{H}_{2} \mathrm{O}(\mathrm{l})\)

03

Addition of manipulated equations & enthalpies

Add the three modified equations: \(\frac{1}{2} \mathrm{N}_{2}(\mathrm{g}) + \frac{3}{2} \mathrm{H}_{2}(\mathrm{g}) - 2 \mathrm{NH}_{3}(\mathrm{g}) + 3 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow \mathrm{NH}_{3}(\mathrm{g}) - 2 \mathrm{NO}(\mathrm{g}) - 3 \mathrm{H}_{2} \mathrm{O}(\mathrm{l}) + 3 \mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \). Cancel terms on both sides to obtain the desired equation. Correspondingly, add the enthalpies with modifications. The overall \(\Delta H = \Delta H_{1}^{\circ} - 2 \Delta H_{2}^{\circ} + 3 \Delta H_{3}^{\circ} \).

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