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Calorimetry is the science of measuring the heat of chemical reactions or physical changes as well as heat capacity. Here, you can think of a calorimeter as a kind of device used to measure amounts of heat exchanged or transformed in a system. Let’s understand the exercise: an insulated beaker can be seen as a simple calorimeter where the water and ice exchange heat without any loss to the surrounding. The main idea in calorimetry is to ensure that all the heat exchange happens between the substances inside the calorimeter.

In our problem, the exercise uses the calorimetry principle to discover how much ice is needed to reduce the hot water's temperature to a specified amount. By keeping track of all the temperature changes and phase transitions, we can maintain a balance of heat exchange, ensuring that the heat lost by the water is equal to the heat gained by the ice.

Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius (or Kelvin). Each material has its own specific heat capacity, meaning they require different amounts of heat to increase their temperature. This property is crucial in our exercise.

We observe two different specific heat capacities in action: one for ice and one for water. For instance:

• The specific heat capacity of ice: 2.09 kJ/kg·K
• The specific heat capacity of water: 4.18 kJ/kg·K

When ice is warming up from (-20.0^{ ext{C}}) to (0^{ ext{C}}), we multiply by the ice's specific heat. When the melted ice (now liquid water) warms up to (40^{ ext{C}}), we use the specific heat capacity of water. This calculation helps us understand exactly how much energy is absorbed at each stage.

Latent heat is the heat required to change the phase of a substance, like melting or freezing, without changing its temperature. It is an essential concept when dealing with phase changes as it involves a large amount of energy, even though the temperature does not change during the transition.

In the exercise, when the ice reaches (0^{ ext{C}}), it needs the latent heat of fusion to turn into water. This is calculated as:

• The latent heat of fusion for ice: 334 kJ/kg

This tells us how much energy is required to convert 1 kg of ice at 0°C to water at the same temperature. The process consumes a lot of heat, contributing significantly to reducing the overall temperature of the system.

Conservation of energy is a fundamental concept stating that energy cannot be created or destroyed in an isolated system. Only converted from one form to another. In our context, this is pivotal in letting us set up the calculation that solves our problem.

In the insulated beaker, the heat lost by the hot water is not escaping to the environment; it’s being transferred to the ice. Therefore, the heat gained by the ice must exactly equal the heat lost by the water, representing a closed and balanced system. Mathematically, this is expressed in the exercise as:

(Q_{1} + Q_{2} + Q_{3} = Q_{ ext{water}}). This equation ensures everything is accounted for, providing a way to solve for the unknown mass of ice needed to achieve the final equilibrium temperature.