Ammonium nitrate dissolves spontaneously and endothermally in water at room temperature. What can you deduce about the sign of \(\Delta S\) for this solution process?

Short Answer

Expert verified
The dissolution of ammonium nitrate in water is spontaneous and endothermic, so \(\Delta G < 0\) and \(\Delta H > 0\). Using the Gibbs free energy equation (\(\Delta G = \Delta H - T \Delta S\)), we deduce that the change in entropy (\(\Delta S\)) must be positive for the inequality to hold true. Therefore, \(\Delta S > 0\) for this solution process.

Step by step solution

01

Recall the Gibbs free energy equation

The Gibbs free energy equation relates the enthalpy, entropy, and temperature of a process to its spontaneity. The equation is given by: \[ \Delta G = \Delta H - T \Delta S\] where \(\Delta G\) is the change in Gibbs free energy, \(\Delta H\) is the change in enthalpy, \(T\) is the temperature in Kelvin, and \(\Delta S\) is the change in entropy. A spontaneous process is characterized by a negative value of \(\Delta G\), i.e., \(\Delta G < 0\).
02

Analyze the given information

We are given that the dissolution of ammonium nitrate is spontaneous and endothermic. This means that \(\Delta G < 0\) and \(\Delta H > 0\) (endothermic implies an increase in enthalpy or a positive value of \(\Delta H\)). From the Gibbs free energy equation, we have: \[\Delta G = \Delta H - T \Delta S < 0\]
03

Deduce the sign of \(\Delta S\)

From the inequality \(\Delta G = \Delta H - T \Delta S < 0\), we have: \[T \Delta S > \Delta H\] Since the temperature \(T\) is positive (room temperature) and \(\Delta H > 0\), we can deduce the sign of \(\Delta S\). For the inequality to hold true, the change in entropy (\(\Delta S\)) must be positive. Thus, the dissolution of ammonium nitrate in water has a positive change in entropy (\(\Delta S > 0\)).

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Most popular questions from this chapter

For each of the following pairs, predict which substance possesses the larger entropy per mole: (a) 1 1 mol of \(\mathrm{O}_{2}(g)\) at \(300^{\circ} \mathrm{C}, 0.01\) atm, or 1 \(\mathrm{mol}\) of \(\mathrm{O}_{3}(g)\) at \(300^{\circ} \mathrm{C}, 0.01\) atm; (b) 1 \(\mathrm{mol}\) of \(\mathrm{H}_{2} \mathrm{O}(g)\) at \(100^{\circ} \mathrm{C}, 1 \mathrm{atm},\) or 1 \(\mathrm{mol}\) of \(\mathrm{H}_{2} \mathrm{O}(l)\) at \(100^{\circ} \mathrm{C}, 1\) atm; \((\mathbf{c}) 0.5 \mathrm{mol}\) of \(\mathrm{N}_{2}(g)\) at \(298 \mathrm{K}, 20 \mathrm{-L}\) volume, or 0.5 \(\mathrm{mol} \mathrm{CH}_{4}(g)\) at \(298 \mathrm{K}, 20-\mathrm{volume} ;(\mathbf{d}) 100 \mathrm{g} \mathrm{Na}_{2} \mathrm{SO}_{4}(s)\) at \(30^{\circ} \mathrm{C}\) or 100 \(\mathrm{g} \mathrm{Na}_{2} \mathrm{SO}_{4}(a q)\) at \(30^{\circ} \mathrm{C} .\)

Classify each of the following reactions as one of the four possible types summarized in Table \(19.3 :\) (i) spontanous at all temperatures; (ii) not spontaneous at any temperature; (iii) spontaneous at low \(T\) but not spontaneous at high \(T ;\) (iv) spontaneous at high T but not spontaneous at low \(T .\) $$ \begin{array}{c}{\text { (a) } \mathrm{N}_{2}(g)+3 \mathrm{F}_{2}(g) \longrightarrow 2 \mathrm{NF}_{3}(g)} \\ {\Delta H^{\circ}=-249 \mathrm{kJ} ; \Delta S^{\circ}=-278 \mathrm{J} / \mathrm{K}}\\\\{\text { (b) } \mathrm{N}_{2}(g)+3 \mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{NCl}_{3}(g)} \\\ {\Delta H^{\circ}=460 \mathrm{kJ} ; \Delta S^{\circ}=-275 \mathrm{J} / \mathrm{K}} \\ {\text { (c) } \mathrm{N}_{2} \mathrm{F}_{4}(g) \longrightarrow 2 \mathrm{NF}_{2}(g)} \\ {\Delta H^{\circ}=85 \mathrm{kJ} ; \Delta S^{\circ}=198 \mathrm{J} / \mathrm{K}}\end{array} $$

Using data from Appendix \(\mathrm{C}\) , calculate \(\Delta G^{\circ}\) for the following reactions. Indicate whether each reaction is spontaneous at 298 \(\mathrm{K}\) under standard conditions. (a) \(2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{SO}_{3}(g)\) (b) \(\mathrm{NO}_{2}(g)+\mathrm{N}_{2} \mathrm{O}(g) \longrightarrow 3 \mathrm{NO}(g)\) (c) \(6 \mathrm{Cl}_{2}(g)+2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s) \rightarrow 4 \mathrm{FeCl}_{3}(s)+3 \mathrm{O}_{2}(g)\) (d) \(\mathrm{SO}_{2}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{S}(s)+2 \mathrm{H}_{2} \mathrm{O}(g)\)

(a) What sign for \(\Delta S\) do you expect when the volume of 0.200 mol of an ideal gas at \(27^{\circ} \mathrm{Cis}\) increased isothermally from an initial volume of 10.0 \(\mathrm{L} ?(\mathbf{b})\) If the final volume is 18.5 \(\mathrm{L}\) , calculate the entropy change for the process. (c) Do you need to specify the temperature to calculate the entropy change?

Which of the following processes are spontaneous: (a) the melting of ice cubes at \(-10^{\circ} \mathrm{C}\) and 1 atm pressure; (b) separating a mixture of \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\) into two separate samples, one that is pure \(\mathrm{N}_{2}\) and one that is pure \(\mathrm{O}_{2}\);(c) alignment of iron filings in a magnetic field; (d) the reaction of hydrogen gas with oxygen gas to form water vapor at room temperature; (e) the dissolution of HCl(g) in water to form concentrated hydrochloric acid?

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