It is well known that when you add salt to ice, the ice not only melts but will actually get colder. From chemistry books, I've learned that salt will lower the freezing point of water. But I’m a little confused as to why it results in a drop in temperature instead of just ending up with water at 0 °C.
What is occurring when salt melts the ice to make the temperature lower?
When you add salt to an ice cube, you end up with an ice cube whose temperature is above its melting point.
This ice cube will do what any ice cube above its melting point will do: it will melt. As it melts, it cools down, since energy is being used to break bonds in the solid state.
(Note that the above point can be confusing if you're new to thinking about phase transitions. An ice cube melting will take up energy, while an ice cube freezing will give off energy. I like to think of it in terms of Le Chatelier's principle: if you need to lower the temperature to freeze an ice cube, this means that the water gives off heat as it freezes.)
The cooling you get, therefore, comes from the fact that some of the bonds in the ice are broken to form water, taking energy with them. The loss of energy from the ice cube is what causes it to cool.
A mixture of water and ice stabilizes at the freezing point of water.
If the ice were any colder, it would absorb heat from the water, in the process raising its own temperature while freezing some part of the water.
If the water is any hotter, it will cool down by melting some of the ice.
This works because ice thawing is endothermic; energy (heat) is used up to turn solid into liquid even though the temperature is staying the same.
The freezing point of water is $0 \pu{°C}$, so water-ice slush stays at $0 \pu{°C}$. If it was lower, it would stabilize at the lower temperature. By adding salt, you are lowering the freezing temperature. The mixture stabilizes there and is colder.
We know that melting or freezing is an equilibrium process. The energy that is required to melt an ice cube will not contribute in elevating its temperature until all the solid water is molten.
If we take two ice cubes and add salt to one of them, then put each of them at room temperature, both of the ice cubes will absorb energy from the surroundings, and this energy as we said will contribute in breaking down the bonds between water molecules.
The cube that salt has not been added to, has a melting point $0~\mathrm{^\circ C}$ and so if we measure its temperature during melting it will remain zero until all ice is molten. That ice cube to which we have added salt, the salt that is added lowers the melting and freezing points of water because it lowers the vapor pressure of water. This ice cube will absorb energy from the environment to help break bonds between water molecules. We know that the salt added will dissolve in the melted portion of the ice. This formed solution of salt will have a lowered freezing point, so the equilibrium between the solid phase and the aqueous phase will be shifted towards the liquid phase since such a solution will freeze at say $-2~\mathrm{^\circ C}$. Since both phases are close together, the ice will absorb energy from the salt solution and will reduce its temperature to the $-2~\mathrm{^\circ C}$ to maintain the equilibrium. When all ice is molten we end up with a salt solution that has got a temperature of say $-1.5~\mathrm{^\circ C}$. This is due to the solution being diluted now. After that, it will start absorbing heat from the room and reach zero and above. So, in conclusion that is how salt melts ice.
When you dissolve $\ce{NaCl}$ in water, it will have to take energy from the system to break its structure so it can dissolve in water. This is the reason the water gets colder because the salt uses the energy from the water to solve it. Now let's look at why ice melts when salt is added. This is based on a so-called colligative attribute. These attributes are only dependent on the amount of substance. When you add particles to a solvent, its vapor pressure lowers. This will result in a higher boiling point(using salt for cooking) and a lower freezing temperature for the solution.
I hope this gives a starting point for further reading consult books on physical chemistry(for e.g. Atkins).
Asking why
When you ask why, you want to know about causality. If I ask "why does the cold pack show a decrease in temperature" and the answer is "because the reaction is endothermic", this might be considered a tautology. After all, endothermic means that energy is needed, and this energy can come from the surrounding, lowering the temperature.
Why does ice water get colder when salt is added?
As the OP states, this lowers the freezing point of the liquid. The system is no longer at equilibrium, and some ice will melt in an endothermic process. As a consequence, the temperature drops and the salt water gets diluted. The melting process stops when salt concentration and temperature are matched again, i.e. the freezing point of the liquid is equal to the temperature of the system.
It is well known that when you add salt to ice, the ice not only melts but will actually get colder.
The melting process is at the interface of liquid and solid, so both the solution and the ice will get colder.
From chemistry books, I've learned that salt will lower the freezing point of water. But I’m a little confused as to why it results in a drop in temperature instead of just ending up with water at 0 °C.
So the question is given that some ice melts, why does the temperature drop. Saying that ice melting is an endothermic process maybe does not fully answer the question (explain the causality).
What is occurring when salt melts the ice to make the temperature lower?
In terms of kinetics, the salt does not melt the ice. Instead, it lowers the rate of water freezing. The net effect is that ice melts. At the molecular level, according to https://www.nyu.edu/pages/mathmol/textbook/info_water.html, "In liquid water each molecule is hydrogen bonded to approximately 3.4 other water molecules. In ice each each molecule is hydrogen bonded to 4 other molecules." So upon melting, water loses about half a hydrogen bond. Also, the remaining hydrogen bonds might have less ideal distances and angles. So that's what makes the process endothermic. The NaCl has little role in the energetics, as any other solute has pretty much the same effect (colligative property).
When you add salt to the ice it melts, I won't go into why since you didn't ask that; all you need to do is that it does if you don't believe me ->
http://science.howstuffworks.com/nature/climate-weather/atmospheric/road-salt.htm
Moving on, whenever a substance undergoes a phase change it's temperature does not rise and usually stays relatively constant, if you looked at a graph of most substances when undergoing different phase changes (i.e. solid to liquid to gas) you will observe regions that are 'flat' or horizontal this is because the energy is no longer causing a rise in temperature but a change in state. Since you have dissolved salt in the ice it will lower the freezing point (note that freezing and melting point of any substance is the same they can be seen as mirrors for one another) this means that water can now exist at lower temperatures and not turn into ice or in other words it will begin to melt at lower temperatures this could attribute as to why the temperature would LOWER as it no longer needs to reach as high a temperature to begin to melt. I haven't explained it very clearly but I hope you understand it consulting a physics and chemistry text book as the person above has suggested is a good idea.
Melting is endothermic and freezing is exothermic. We never observe water warm up when it freezes because more energy has to be lost from the system before more water freezes. When water freezes from being in cold air, the release of heat actually slows down the freezing. When you add salt to a mixture of water and ice, it causes more ice to melt by depressing the freezing point and not by adding internal energy so it actually gets colder.
Let’s do a thought experiment to see what is going on. Combine 500 mL of pure H2O and 500 grams of ice, each at 0 C, in a perfectly insulated container. Then add 300 grams of NaCl (also at 0 C). At first, not all the NaCl will dissolve, because NaCl is only soluble 26% in water at 0 C. (The temperature of the water will decrease slightly because of the energy required to dissolve the salt: about 1 kcal/mole of salt.)
The melting point of the ice cube is still 0 C. However, the liquid surrounding it, although at approximately 0 C, is a near-saturated salt solution, not pure water.
Water molecules from the ice cube will diffuse into the salt solution, diluting it. Energy is required for the transition from solid ice to water: the amount of heat required to melt ice is 68.3 kcal/mole (heat of fusion). This energy comes from the salt solution, reducing its temperature. The ice cube will continue to melt (turns from solid to liquid) as long as it is not in equilibrium with the solution. The ice cube does not melt because the surrounding liquid has a lower freezing point, it melts because the surrounding solution is less than 100% water. The surrounding solution is already colder than 0C and the ice cube is still pure solid water at 0C. In other words, the driving force is a non-equilibrium of concentration. At the beginning of the experiment, there was thermal equilibrium (everything at 0C), but a concentration non-equilibrium forced the melting of ice by heat energy from the solution. Freezing point depression characterizes a solution, but is not a driving force.
No. Unfortunately we seem to all be forgetting a fundamental property of ice - I kicked myself when I realized it. Think of a large chunk of ice taken out of your home freezer, sitting in a bucket, floating in its own melt-water. All the liquid water surrounding that ice is at $\pu{0 ^\circ C}$. The extreme surface of that ice, that is exposed to the meltwater, (the layer that is about to transition to liquid) is also at $\pu{0 ^\circ C}$. BUT the solid (non-transitioning) interior of the ice chunk is still at about the temperature of the freezer that it came from, about $\pu{-18 ^\circ C}$! So we shouldn’t think of all the water in the bucket (liquid & solid) as being at $\pu{0 ^\circ C}$. Instead, most of the frozen water is much colder than $\pu{0 ^\circ C}$. So now we see that within the chunk of ice there is a significant temperature gradient. The exact center is coldest (at perhaps $\pu{-18 ^\circ C}$). Moving out from there the temperature increases until you reach the surface where the ice has warmed to a temperature of $\pu{0 ^\circ C}$ and has begun melting. Now we can readily see why adding salt to the liquid surrounding the ice chunk would cause a lowering of the temperature of that salty melt-water. By adding salt you have lowered the melting point. The surface layer of the ice that was $\pu{0 ^\circ C}$ would rapidly melt because the salt is in close contact with it and acts on it such that the ice surface is now much too warm to maintain the solid state. Underneath that quickly melting ice layer is another layer at say $\pu{-1 ^\circ C}$. The salt would be able to act on that colder ice in turn and cause it also to melt while maintaining that negative temperature creating a small amount of $\pu{-1 ^\circ C}$ briny water. This below-zero melting action caused by the briny liquid would continue toward a new temperature equilibrium (between the solid surface and the liquid) somewhere below $\pu{0 ^\circ C}$ in accordance with the salinity level of the melt-water.
