91Ó°ÊÓ

Heat exchange is a fundamental concept in thermodynamics, describing the transfer of thermal energy between substances. In the context of our problem, we are dealing with the heat exchange between ice and water. When substances at different temperatures come into contact, heat will flow from the hotter substance to the colder one until they reach the same temperature. This process obeys the principle of conservation of energy: energy lost by the hot substance must be equal to energy gained by the cold one. The math of this exchange relies on key formulas: for instance, the heat lost or gained by a substance can be calculated using the equation \[ Q = mc\Delta T \], where \[ Q \] is the heat exchanged, \[ m \] is the mass, \[ c \] is the specific heat capacity, and \[ \Delta T \] is the change in temperature.

Latent heat of fusion refers to the amount of energy required to change a substance from solid to liquid at its melting point without changing its temperature. In our problem, we need to melt ice at 0°C, so we use the latent heat of fusion for ice, which is 334 J/g. This means that to melt 100 g of ice, we need \[ Q_{melt} = m_{ice} \times L_{f} = 100 \textg \times 334 \textJ/g = 33400 \textJ \]. It is essential to understand that this energy is only used to change the state of the ice and does not increase its temperature. The melted ice will remain at 0°C until all the ice has turned into liquid water.

Specific heat capacity is a measure of how much heat energy is required to raise the temperature of one gram of a substance by one degree Celsius. Different substances have different specific heat capacities. Water has a specific heat capacity of 4.18 J/g°C. In our problem, to find how much heat the water at 80°C loses when it cools down to 0°C, we use the formula \[ Q_{water} = m_{water} \times c_{water} \times \Delta T = 100 \textg \times 4.18 \textJ/g°C \times (80 - 0)°\textC = 33440 \textJ \]. This calculation helps us understand how much heat is available to melt the ice and what happens to the temperature of the water.

Thermal equilibrium is the state in which all parts of a system have reached the same temperature and no more heat flows between them. In our context, we are looking to find the equilibrium temperature after mixing ice and water at different initial temperatures. Once the ice melts with the heat provided by the warmer water, both substances start to exchange heat until they reach a common temperature. After melting 100g of ice with 33400 J, the leftover 40 J warms up the resulting water. Using the specific heat capacity for water, we find the final temperature increase \[ 40 = 200 \times 4.18 \times \Delta T \] which simplifies to \[ \Delta T = \frac{40}{200 \times 4.18} = 0.048^{\circ}C \]. Adding this to the melting point of ice (0°C), we get a final equilibrium temperature of approximately 0.048°C.