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Specific heat capacity is a fundamental concept in thermodynamics that describes how much heat energy a substance can absorb before it changes temperature. It's like a measure of how "heat retentive" a material is. Each material has its specific heat capacity, denoted by the symbol \( c \), and this value tells us how much energy (in joules) is required to raise the temperature of one kilogram of the substance by one degree Celsius.
In our exercise, the coffee mug and water both require heat to increase their temperatures from \( 15^{\circ} \mathrm{C} \) to \( 100^{\circ} \mathrm{C} \). For the mug, its specific heat capacity is \( 920 \mathrm{J} / (\mathrm{kg} \cdot \mathrm{C}^{\circ}) \). This value is significant in calculating the amount of energy needed for heating. The higher the specific heat capacity, the more energy you'll need to change the temperature.

• High Capacity: Substances like water, which have a high specific heat capacity (\( 4186 \mathrm{J} / (\mathrm{kg} \cdot \mathrm{C}^{\circ}) \)), absorb more energy without a big temperature change. This makes them excellent for storing and transporting heat.
• Low Capacity: Conversely, metals typically have low specific heat capacities, meaning they heat up and cool down quickly.

The specific heat capacity plays a key role in determining how different substances will react to heat energy — ensuring accurate thermal management in practical applications.

Calculating power is about understanding the rate at which energy is used. In physics, power is defined as the amount of energy transferred or converted per unit of time. The formula for power is \( P = \frac{Q}{t} \), where \( P \) is the power in watts, \( Q \) is the energy in joules, and \( t \) is the time in seconds.
Using the exercise, we need to know how much power the heater must have to bring the coffee mug and water to a boil in three minutes (or 180 seconds). We already calculated the total heat energy needed, \( Q_{\text{total}} = 116362.5 \mathrm{J} \). Now, we're figuring out how quickly that energy must be supplied:

• Power Calculation: Substitute the values into the formula: \( P = \frac{116362.5}{180} \).
• Result: The calculated power is \( 646.46 \mathrm{W} \). This means the heater needs a minimum power rating of \( 646.46 \mathrm{W} \) to efficiently perform the task within the given time.

Understanding power calculations helps us design systems that effectively balance energy efficiency and performance — crucial for everything from household appliances to industrial machinery.

Temperature change is an essential concept that shows how the temperature of a substance varies when heat energy is applied. In thermodynamics, this is indicated by \( \Delta T \), which is the difference between the final and initial temperatures of a substance.
In the coffee mug scenario, we're interested in changing the temperature from \( 15^{\circ} \mathrm{C} \) to the boiling point of water, \( 100^{\circ} \mathrm{C} \), resulting in a temperature change of \( 85^{\circ} \mathrm{C} \). This change helps us understand how much energy is needed to achieve this transformation.
Let's break down why it's crucial:

• Energy Requirement: The greater the temperature change, the greater the amount of energy required. This is directly used in the formula \( Q = mc\Delta T \), where \( \Delta T \) significantly impacts the heat needed.
• Practical Implications: Knowing \( \Delta T \) allows us to plan heating or cooling processes effectively. For instance, we need more energy to achieve larger temperature changes in heating applications.

Temperature change isn't just a number; it's a key parameter influencing how we manage energy resources in both daily life and advanced technological settings.