91Ó°ÊÓ

Specific heat capacity, often simply called specific heat, is a property of materials that tells us how much heat energy is required to change the temperature of a given mass. It is defined as the heat required to raise the temperature of one kilogram of a material by one degree Celsius. The formula we use is:

• \[Q = mc\Delta T\]

In this formula, \( Q \) represents the heat energy (in joules or kilojoules), \( m \) stands for mass (in kilograms), \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature.

Different materials have different specific heat capacities. For example, steel has a much lower specific heat than wood. This means that, for the same mass and temperature change, steel requires less energy to heat up compared to wood. In our problem, the specific heat capacity for steel is \( 0.46 \text{ kJ/kg°C} \) and for wood, it's \( 2.3 \text{ kJ/kg°C} \). These values significantly affect the calculations for how much heat is needed to increase the temperature of the houses.

Heat transfer is the process of thermal energy moving from a warmer object to a cooler one. In the context of heating homes, it's crucial to understand how heat is conducted through materials. We often use furnaces to provide a consistent source of heat energy, which transfers thermal energy evenly throughout a space.

• Heat naturally moves from warmer to cooler areas.
• The rate of heat transfer is influenced by material properties, the temperature difference, and the surface area involved.

In our problem, each house receives heat from a furnace outputting \( 10^5 \text{ Btu/hr} \), equivalent to \( 105500 \text{ kJ/hr} \). Understanding the relationship between heat input and the specific characteristics of materials is essential for efficient heating. The efficient transfer of this energy determines how quickly each house reaches the desired temperature of \( 18^{\circ}\text{C} \).

By calculating the energy required to heat each house, using their specific heat capacities, we understand that both houses require the same amount of heat energy due to the particular characteristics of their materials. However, the method and speed of heat transfer could differ if factored against insulation or the way different materials like wood and steel absorb heat.

Temperature change, denoted as \( \Delta T \), refers to the difference in temperature between the initial state and the final state. In our exercise, the temperature change for each house is from \( 2^{\circ}C \) to \( 18^{\circ}C \).

Calculating \( \Delta T \) is a straightforward process:

• The initial temperature (let's say, \( T_1 \)) is \( 2^{\circ}C \).
• The final temperature (\( T_2 \)) desired is \( 18^{\circ}C \).
• The temperature change \( \Delta T \) is simply \( T_2 - T_1 \).
• In our case, \( \Delta T = 18^{\circ}C - 2^{\circ}C = 16^{\circ}C \).

This change helps us calculate the total heat energy needed to warm a particular material. Since both houses have the same \(\Delta T\), we can focus on differences in their mass and specific heat to understand their heating requirements.