91Ó°ÊÓ

In order to determine the sign of \(\Delta S\) for this process, we need to recall the relationship between enthalpy, entropy, and Gibbs free energy. The Gibbs free energy is an important thermodynamic quantity that indicates whether a process will occur spontaneously. The change in Gibbs free energy (∆G) for a process can be calculated using the following equation: \[ \Delta G = \Delta H - T\Delta S \] Where: - \(\Delta G\) is the change in Gibbs free energy - \(\Delta H\) is the change in enthalpy (heat content) - \(T\) is the temperature in Kelvin - \(\Delta S\) is the change in entropy (dispersal of matter and energy) A negative value for \(\Delta G\) means that the process occurs spontaneously, while a positive value means the process is non-spontaneous.

We are given that the dissolution of ammonium nitrate in water is an endothermic process. By definition, an endothermic process means that it absorbs heat, and so the enthalpy change, \(\Delta H\), is positive.

We are also given that the dissolution is spontaneous. This means that the Gibbs free energy change, \(\Delta G\), must be negative.

Finally, we are ready to determine the sign of entropy change, \(\Delta S\). We can rearrange the Gibbs free energy equation: \[ \Delta G = \Delta H - T\Delta S \] Since we know that \(\Delta G < 0\) (spontaneous) and \(\Delta H > 0\) (endothermic), the only way to satisfy the equation is if \(\Delta S\) is positive: \[ \Delta G = (+) - T(+) \implies (-) = (-)\] In other words, the positive entropy change \(\Delta S\) compensates for the positive enthalpy change \(\Delta H\) to make the dissolution of ammonium nitrate a spontaneous process.

The entropy change, \(\Delta S\), for the dissolution of ammonium nitrate in water is positive. This positive entropy change occurs because the dissolution of ammonium nitrate leads to an increase in disorder in the system as a result of the dispersal of matter and energy.