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Understanding specific heat capacity is crucial when assessing energy changes in materials like water. Specific heat capacity, denoted by the symbol \( c \), is a measure of the amount of heat energy needed to change the temperature of a given mass of a substance by one degree Celsius.
In this example, the specific heat capacity of water is \( 4186 \, \text{J/kg}\cdot{^{\circ}C} \). This means that to raise the temperature of 1 kg of water by 1掳C, you require 4186 Joules of energy.
To determine the energy released or absorbed when water changes temperature, we use the formula:

• \( Q = mc\Delta T \)

You multiply the mass \( m\) in kilograms by the specific heat capacity \( c \) and the change in temperature \( \Delta T \). If water cools from \( 10.0^{\circ} \text{C} \) to \( 0.0^{\circ} \text{C} \), the energy released is \( 27,627,600 \, \text{J} \).
This calculation is essential for understanding how much energy is being transferred as heat when the water cools.

When water transitions from a liquid to a solid state (or vice versa), it involves latent heat. Latent heat of fusion, specifically, refers to the energy required to change a unit mass of a substance from liquid to solid at its melting point, without changing its temperature.
For water, the latent heat of fusion is \( 334,000 \, \text{J/kg} \).
When 660 kg of water at \( 0.0^{\circ} \text{C} \) turns into ice, the energy released is calculated using:

• \( Q = mL \)

Where \( m \) is mass and \( L \) is the latent heat of fusion. In our scenario, this energy release amounts to \( 220,440,000 \, \text{J} \). This is a significant portion of the total energy and highlights why transitions involving phase changes are potent sources of energy storage and release without temperature variation.

Energy calculation involves summing different contributions of energy changes. In thermodynamics, knowing the total energy change, whether absorbed or released, allows us to understand the system's dynamics.
In the greenhouse example, you first calculate the energy released by the sensible heat loss (cooling water), and then add the latent heat released by the freezing process.
Here are the steps involved:

• Calculate energy from cooling: \( Q_1 = mc\Delta T \)
• Calculate energy from phase change: \( Q_2 = mL \)
• Total Energy: \( Q_{\text{total}} = Q_1 + Q_2 \)

This approach allows a simplified yet comprehensive view of the energy transformations occurring as temperature and phase changes are interrelated. For this scenario, we end up needing \( 248,067,600 \, \text{J} \) of energy to achieve the same effects artificially.

An electric heating system can be relied upon to provide heat if the natural freezing process isn鈥檛 what is desired. Calculating the minimum requirement for such a system involves using the principles from energy and power calculations.
The power is calculated based on the energy it should provide over a certain period. In this problem, the total energy over nine hours (converted to seconds) needs to be managed, calculating the required power using:

• \( P = \frac{Q_{\text{total}}}{\text{time}} \)
• Where 9 hours = 32,400 seconds

With a calculated power need of approximately \( 7655 \, \text{W} \), you then determine the minimum current, using the power equation: \( P = IV \), where \( V = 240 \, \text{V} \). Solving for \( I\), you find the system needs to handle about \( 31.9 \, \text{A} \).
Understanding this requirement is key to ensuring efficiency and safety of the heating system.