Question

An instant cold pack purchased in a pharmacy contains a packet of solid ammonium nitrate surrounded by a pouch of water. When the packet of \(\mathrm{NH}_{4} \mathrm{NO}_{3}\) is broken, the solid dissolves in water and a cooling of the mixture occurs because the solution process for \(\mathrm{NH}_{4} \mathrm{NO}_{3}\) in water is endothermic. Explain, in terms of what happens to the molecules and ions, why this mixing occurs spontaneously.

Short answer

The spontaneous mixing of solid ammonium nitrate and water is due to the increase in entropy as ions dissolve and disperse, which outweighs the endothermic enthalpy change, leading to a decrease in free energy of the system.

Step-by-step solution

Understanding Spontaneity in Chemical Reactions

To understand why the mixing occurs spontaneously despite being endothermic, it is necessary to consider the thermodynamic concept of free energy. A spontaneous process is one that results in a decrease in the free energy of a system. The free energy change (∆G) can be broken down into two components: the enthalpy change (∆H) and the entropy change (∆S), following the equation ∆G = ∆H - T∆S, where T is the temperature.

Analyze Enthalpy Change (∆H)

The process is endothermic, meaning that the enthalpy (∆H) is positive. This implies that energy is absorbed from the surroundings to dissolve the ammonium nitrate (NH_{4}NO_{3}). However, an endothermic process can still be spontaneous if it is accompanied by an increase in entropy that outweighs the enthalpy change.

Examine Entropy Change (∆S)

When solid ammonium nitrate dissolves in water, the ions of NH_{4}^{+} and NO_{3}^{-} are released into solution, increasing the entropy (∆S) of the system because the ions are more dispersed in the solution than in the solid state. The water molecules also become more ordered around the ions, which contributes to a further increase in entropy.

Consider Temperature's Role in Spontaneity

At a given temperature, if the entropy term (T∆S) is sufficiently large to overcome the increase in enthalpy, then the free energy change (∆G) will be negative, indicating a spontaneous process. Since the dissolution of ammonium nitrate is spontaneous, it is clear that the entropy factor is significant and results in a decrease in free energy.

Key concepts

Free Energy in Chemical Reactions

When we try to understand whether a chemical reaction will happen by itself—without any outside help—we talk about the concept of 'free energy'. This is not the kind of energy you get from a snack or a nap; instead, it's a special type of energy in chemistry that tells us if a reaction wants to happen on its own. Imagine you have a skateboard at the top of a ramp. If the skateboard rolls down the ramp, that's like a spontaneous reaction—it just happens. The skateboard had 'free energy' to start with, and as it went down, it used up that energy.

In chemistry, we calculate this energy with a nifty equation: \( \Delta G = \Delta H - T\Delta S \), where \( \Delta G \) is the free energy change, \( \Delta H \) is the enthalpy change (like the energy from a warm cookie), \( \Delta S \) is the entropy change (how messy your room is), and \( T \) is the temperature, measured in Kelvin. If \( \Delta G \) is negative, it's like the skateboard rolling down; the reaction will happen without being pushed. So even when a chemical reaction is like climbing up the ramp—basking in energy from outside like our endothermic friend, ammonium nitrate—it can still roll down spontaneously if it's really good at getting messy (increasing entropy)!

Enthalpy Change

Enthalpy change (\(\Delta H\)) is a fancy term scientists use to describe the heat energy change in a reaction. Think of it like your body warming up when you munch on a tasty snack—the food gives you energy. When a reaction has a negative \(\Delta H\), it gives off heat, kind of like you feel warm on a sunny day. But sometimes reactions are the opposite—they absorb heat, like how an ice pack gets cold. That's what's happening with the cold pack that contains ammonium nitrate; it's taking in heat from its surroundings because its \(\Delta H\) is positive, which feels cold to us.

It might seem odd that a cold pack can get cold by itself, but it's all because of the balance—sometimes absorbing heat is just part of the reaction's journey to becoming spontaneous, like putting in the effort to ride a bike uphill before zooming down the other side with the wind in your hair.

Entropy Change

Entropy (\(\Delta S\)) is how scientists measure the messiness or randomness in a system. It's like if your toys are spread all over your room—it's high entropy. But if they're all neatly packed away—in low entropy. Now, in the chemical world, we can think of solid ammonium nitrate as a well-organized toy box. When it dissolves in water, those ammonium (\(NH_4^+\)) and nitrate (\(NO_3^-\)) ions spread out and mix with the water molecules, creating a bigger mess. This messiness helps reactions happen spontaneously because the universe sort of likes it when things spread out and get a bit chaotic.

Even if we need to put in energy to get the mess started (like the endothermic reaction we're talking about), as long as the final messiness (\(\Delta S\)) is really big, it can make the whole process happen on its own—the messiness wins!

Dissolution of Ammonium Nitrate

The dissolution of ammonium nitrate (\(NH_4NO_3\)) in water is like magic in a bag! It's an endothermic reaction, which means it's like a sponge soaking up water. But instead of water, it's soaking up heat from its environment—yep, it's getting energy from the water and the air around it, and that's why it feels cold to the touch.

As the solid ammonium nitrate dissolves, it breaks into \(NH_4^+\) and \(NO_3^-\) ions, which mingle with the water molecules. The water molecules love these ions, so they get all organized around them, increasing the messiness (or entropy), and remember, the universe loves a good mess. Because of this increase in entropy, even though the reaction needs heat to get going (positive \(\Delta H\)), it still happens naturally—spontaneously! It's like investing some energy upfront to throw a great party that everyone talks about because everyone spread out and had a great time!