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Specific heat capacity is a property that tells us how much heat energy is required to raise the temperature of a certain mass of a substance by one degree Celsius. In this exercise, we dealt with two specific values for the substances involved:

• Coffee mug: 920 \( J/(kg \cdot C^{\circ}) \)
• Water: 4186 \( J/(kg \cdot C^{\circ}) \)

The large specific heat capacity of water means it requires more energy to heat compared to the material of the coffee mug. Knowing this helps us calculate the energy required to bring both the mug and water to a desired temperature.
The concept of specific heat capacity is vital in thermodynamics as it determines how substances react to energy changes.

Energy transfer in this context refers to the process of transferring heat from the heater to both the coffee mug and water. The formula used to determine this transfer is:\[ Q = m \cdot c \cdot \Delta T \]Where:

• \( Q \) is the heat energy transferred
• \( m \) is the mass
• \( c \) is the specific heat capacity
• \( \Delta T \) is the change in temperature

This equation allows us to find out how much heat energy is specifically required to bring the mug and water from 15°C to 100°C. By carefully calculating \( Q \) for both the water and the mug independently, and then summing them up, we get the total energy required for the entire heating process.

Calculating power involves determining how quickly energy is transferred or used. Power is given by the formula:\[ P = \frac{Q_{total}}{time} \]Here:

• \( P \) is the power
• \( Q_{total} \) is the total energy required
• \( time \) is the duration of energy transfer

In this exercise, the total energy needed to bring the water and mug to a boil is divided by the time taken (180 seconds). The calculated power helps us determine the minimum power rating of the heater required to achieve this within the given timeframe. The real-world application of this concept allows us to design appliances like heaters efficiently.

Temperature change is a crucial aspect of solving problems in thermodynamics. In this exercise, the change in temperature \( \Delta T \) was calculated as the final temperature minus the initial temperature:\[ \Delta T = T_f - T_i = 100^{\circ}C - 15^{\circ}C = 85^{\circ}C \]This means that both the coffee mug and the water need to be heated by 85°C.

The change in temperature is directly related to the amount of energy required to heat an object. Knowing the specific starting and ending temperatures allows us to precisely determine the energy demands of the process. This fundamental calculation underpins many practical applications, ensuring we use energy efficiently and effectively.