Ice at \(0^{\circ} \mathrm{C}\) is placed in a Styrofoam cup containing \(361 \mathrm{~g}\) of a soft drink at \(23^{\circ} \mathrm{C}\). The specific heat of the drink is about the same as that of water. Some ice remains after the ice and soft drink reach an equilibrium temperature of \(0^{\circ} \mathrm{C}\). Determine the mass of ice that has melted. Ignore the heat capacity of the cup. (Hint: It takes 334 J to melt \(1 \mathrm{~g}\) of ice at \(\left.0^{\circ} \mathrm{C} .\right)\)

Understanding how thermal energy, or heat, moves from one object to another is essential in studying various scientific and engineering disciplines. Heat transfer is the physical act of thermal energy being exchanged between two systems or a system and its surroundings. In our exercise, heat transfer occurs between the ice and the soft drink when they are combined in the Styrofoam cup. When the warmer soft drink at 23°C comes into contact with the 0°C ice, heat flows from the soft drink to the ice until they reach thermal equilibrium at 0°C.

The heat transfer is governed by the law of conservation of energy, which states that energy cannot be created or destroyed, only transferred. Therefore, the heat lost by the soft drink is the same as the heat gained by the ice. The transfer of heat continues until the temperatures equalize, at which point thermal equilibrium is achieved. This is a practical example of heat transfer where the critical element is the concept of energy balance. In solving such problems, it's important to consider the direction of heat flow and the end state of the system.

The term specific heat capacity refers to the amount of heat needed to raise the temperature of one gram of a substance by one degree Celsius (or one kelvin). It is a property intrinsic to the material and provides insight into how different substances react to the addition or loss of heat. An interesting example is the specific heat capacity of water (and substances with similar specific heat capacities), like the soft drink in our exercise, which is relatively high at 4.18 J/g°C.

In practical applications, the high specific heat capacity of water implies that it can absorb or release a large amount of heat with minimal temperature change. This characteristic is crucial for understanding the energy exchange within the exercise. It is significant to note that due to the high specific heat capacity, water and the soft drink are effective at absorbing the energy from the ice without a significant increase in their temperatures, which enables the melting process of the ice in our example.

The field of thermodynamics involves the study of energy transformations and the relationships among properties of substances, particularly relating to heat and work. In the context of our exercise, the principles of thermodynamics govern the behavior of the soft drink and ice as they come into contact and exchange energy. The first law of thermodynamics, also known as the law of energy conservation, plays a central role in this scenario, asserting that the energy cannot be created or destroyed in an isolated system.

The thermodynamic process that occurs when ice at 0°C absorbs heat without changing temperature until it reaches its melting point is an example of a phase change. This phenomenon is related to the concept of latent heat, which for ice is called the latent heat of fusion. This heat is absorbed without a temperature rise as it is used to break the bonds between water molecules, transitioning them from a solid to a liquid state. The exercise illustrates a thermodynamic process where the energy lost by the soft drink (through heat transfer) is precisely the energy gained by the ice, leading to a phase change instead of a temperature change.