Thermal equilibrium: Thermal energy of the system is equal to thermal energy of the surroundings. Heat is defined as the energy transfer resulting from differences in thermal energy. Heat always flows from higher temperature to lower temperature. Heat transfer between a system and its surroundings stop when they reach thermal equilibrium, or when there is no difference in thermal energy. In this case, ice was dropped into the cup. Initially, the ice increases the temperature of the water, creating a cold glass with condensation. However, energy from the surroundings flow into the system (the glass of water) due to thermal difference and warm the glass of water until the two reach thermal equilibrium. At thermal equilibrium, there is no ice, no condensation, and the water temperature is room temperature.   

In this question, we're told that a certain amount of a compound dissolved in water lowers the water's freezing point. We're asked to determine the molar mass of the compound.

First, we must recognize this as a freezing point depression problem. Recall that freezing point depression is one of the colligative properties associated with dissolving solute into a solvent such as water.

To begin, we need to use the equation for freezing point depression, which states that the change in freezing point is proportional to both the molality of the solution as well as the molal freezing point depression constant for the solvent in question (in this case, water).

The answer we have just calculated above is the molality, which is a different way of expressing concentration than molarity. Molality is expressed as the number of moles of solute per kilogram of solvent, whereas molarity is the number of moles of solute divided by the total volume of the solution.

The next thing we need to do is find out how many total moles of solute are present in the solvent water. To do this, we need to use the volume of water provided to us in the question, along with the density of water, to calculate the mass of water present. Together with the molality calculated above, this will allow us to know the total number of moles in solution.

Now that we know the total number of moles of solute (our compound) that exists in solution, we can use that information, together with the total mass of the compound given to us in the question stem, to calculate the compound's molar mass.

Recall that the heat gained by ice must be equal to the heat from the food.

Start by calculating the heat from the food by converting the Calories into joules.

Next, calculate the heat gained by the ice. Take this in three steps:

1. Heating the ice from  to 

2. Ice melting.

3. Raising the temperature of water from  to .

Set this value equal to the heat from the food and solve for the mass.

Convert to kilograms.

Remember that when any substance is undergoing a phase change, its temperature will remain constant. In other words, the energy being added is not increasing the average kinetic energy of any of the particles in the system.

Also, the internal energy will not remain constant. As heat energy is added, as in the question, the internal energy will necessarily increase. Even though the kinetic energy of the particles is not increasing, the potential energy is. This is because the intermolecular forces of attraction (mostly hydrogen bonds in this case) need to be broken apart. When energy is added, that raises the potential energy component of internal energy, despite the fact that kinetic energy (and thus, temperature) remains constant.