For this problem, use Boyle's Law: 

Boyle's Law allows us to set up a proportion between the pressure and volume at a constant temperature.

Using the values given, we can solve for the final pressure.

The volume of air in the balloon will increase when exposed to hotter temperatures, and decrease when exposed to colder temperatures. If we look at the ideal gas law, we can see that temperature and volume have a direct relationship. As one goes down, so does the other, assuming all other factors remain constant.

We can also look at Charles's law of volumes:

The balloon is sealed, so the amount of gas in the balloon will not change, and the elasticity of the balloon means that pressure will also remain constant. As temperature decreases, volume must also decrease. Suppose that the temperature is halved in our question. The result would be half the volume, according to Charles's law.

By this logic, we can conclude that the balloon will shrink when placed in the cold water.

Heat is a form of energy. Adding heat to a gaseous system will increase the energy of the molecules, causing them to move faster and collide more frequently. This increased velocity results in the expansion of the gas.

The formula for the net energy added to a system is;

The change in energy equals the heat energy minus the work done.

We don't know the work, but we can solve for it using the work equation:

We are given both the force and the distance, allowing us to calculate the work.

Now that we know both the work done and the heat added, we can solve for the final energy of the system.

The most likely explanation is that work is done by the system.

The formula for change in energy shows that the net change in energy is equal to the increase in heat energy minus the work done:

Since , there must have been work done by the system.

Heat is a form of energy, while temperature is a measure of the average kinetic energy of the molecules present in a system. Since both systems are measure to be at , their average kinetic energies are the same. The cup and the bathtub have the same temperature; however, since the bathtub contains more water, it contains more molecules. Temperature is the measure of heat energy per molecule. A greater number of molecules at the same temperature is indicative of more heat energy than fewer molecules at that temperature. Since the bathtub has more molecules, it has more heat energy even though the two systems have the same temperature.

A temperature gradient is always needed for heat transfer to occur. The temperature difference is what drives a flow of heat, as heat will always travel from an area of higher temperature to an area of lower temperature. This can occur between two materials, or within a single material. For example, if an iron pot is placed on a stovetop, the entire metal pot will become hot even though only the bottom is in contact with the heat source. This is because the heat transfers through the metal, from the high heat at the bottom to the lower heat at the top.

Enthalpy, or , is the total energy of a thermodynamic system. Similar to how mechanical energy can change during mechanical processes, involving changing distances of velocities, enthalpy will increase or decrease with changes made to the thermodynamic state of the system. It is simply a measure for a different form of energy.

When an object is changing forms (solid to liquid in this case), the temperature remains constant. All of the energy that would normally go towards changing the internal temperature of the object is going into  the latent heat of fusion or enthalpy of fusion instead.

Entropy is the measurement of disorder within a system, or how far it is from thermal equilibrium. Remember that everything in nature tends towards an equilibrium. The further from that equilibrium something is, the more "disordered" it is when compared to nature's preferred state.