Question

Let's say you come home from a winter vacation to find your house at \(5^{\circ} \mathrm{C}\) and you want to heat it to \(20^{\circ} \mathrm{C}\). Let's say the house contains: \(500 \mathrm{~kg}\) of air; \(^{62} 1,000 \mathrm{~kg}\) of furniture, books, and other possessions; plus walls and ceiling and floor that amount to \(6,000 \mathrm{~kg}\) of effective \(^{63}\) mass. Using the catch-all specific heat capacity for all of this stuff, how much energy will it take, and how long to heat it up at a rate of \(10 \mathrm{~kW}\) ? Express in useful, intuitive units, and feel free to round, since it's an estimate, anyway.

Short answer

Answer: The time required to heat the house is directly proportional to the specific heat capacity of the materials in the house. The greater the specific heat capacity, the more time it would take to heat the house, as it would require more energy to raise the temperature.

Step-by-step solution

Identify the given data

We are given: - Initial temperature: \(T_1 = 5^\circ \mathrm{C}\) - Final temperature: \(T_2 = 20^\circ \mathrm{C}\) - Mass of air: \(m_\text{air} = 500 \mathrm{~kg}\) - Mass of furniture, books, and other possessions: \(m_\text{furniture} = 1,000 \mathrm{~kg}\) - Effective mass of walls, ceiling, and floor: \(m_\text{walls} = 6,000 \mathrm{~kg}\) - Power of the heating system: \(P = 10 \mathrm{~kW}\) (which converts to \(10,000 \mathrm{~W}\))

Calculate the temperature difference

We need to find the difference between the initial and final temperatures. \(\Delta T = T_2 - T_1 = 20^\circ \mathrm{C} - 5^\circ \mathrm{C} = 15^\circ \mathrm{C}\)

Calculate the energy required to heat the house

We can use the formula \(Q = mc\Delta T\) to find the energy required to heat the house, where Q is the energy required, m is the mass, c is the specific heat capacity, and \(\Delta T\) is the temperature difference. The total energy required will be the sum of the energies required for each component (air, furniture, and walls). For simplicity, we will use a catch-all specific heat capacity (c) value for all the components in the house, and then sum the masses together. First, we find the total mass: \(m_\text{total} = m_\text{air} + m_\text{furniture} + m_\text{walls} = 500 \mathrm{~kg} + 1,000 \mathrm{~kg} + 6,000 \mathrm{~kg} = 7,500 \mathrm{~kg}\) Next, we calculate the energy needed: \(Q = m_\text{total}c\Delta T = 7,500 \mathrm{~kg} \cdot c \cdot 15^\circ \mathrm{C}\) Note that the energy required will be in joules (J). Without a specific value for c, we can only express the energy required as a function of the specific heat capacity.

Find the time required to heat the house

We can find the time required to heat the house with the given power of the heating system using the formula: \(t =\frac{Q}{P}\). \(t = \frac{7,500 \mathrm{~kg} \cdot c \cdot 15^\circ \mathrm{C}}{10,000 \mathrm{~W}}\) Since the energy required depends on the specific heat capacity, so does the time it takes to heat the house. Once a specific heat capacity value for the materials in the house is given (usually found through research or in a reference table), we can calculate the exact time required to heat the house. The resulting time would be expressed in seconds, but it is more useful and intuitive to convert it to minutes or even hours for practical use. To convert from seconds to minutes, divide by 60, and to convert to hours, divide by 3600 (60 minutes * 60 seconds).

Key concepts

Heat Transfer

Heat transfer is a crucial concept in thermodynamics that explains how thermal energy moves from one substance, or object, to another. In the context of heating a house, heat transfer is the process that ensures the inside temperature rises from a chilly state to a comfortable level.
It often involves:

• Conduction: Direct heat passing through materials, such as from a heater through the walls and air inside a home.
• Convection: Heat transfer through fluids like air or water, which play a significant role in equalizing temperatures within rooms.
• Radiation: Heat moving in the form of electromagnetic waves, contributing to warming up large spaces.

Each of these methods must work together efficiently to increase the temperature in a setting like a house. Thus, understanding heat transfer helps in determining how fast a room heats up and how much energy it will consume.

Specific Heat Capacity

Specific heat capacity is another fundamental concept when discussing energy and heat. It is a measure of the amount of heat per unit mass required to raise the temperature of a substance by one degree Celsius.
This value varies for different materials and plays a critical role in predicting how much energy is necessary to change their temperature.
Key aspects include:

• The higher the specific heat capacity, the more energy is needed to increase the temperature of the material, meaning it warms up gradually but retains heat well.
• Common materials, like water and air, have specific heat capacities that are often used in calculations for practical purposes.
• In our exercise, a universal specific heat capacity is assumed for simplicity when calculating how much energy is needed for heating the house.

An accurate specific heat capacity aids in creating effective energy management, reducing waste and improving efficiency in real-world applications.

Energy Calculation

Calculating energy, especially in the context of heating, requires understanding how much energy (in joules) you need to achieve a desired temperature change. The formula for energy calculation is:\[ Q = mc\Delta T \]where \(Q\) is the heat energy, \(m\) is the mass, \(c\) is the specific heat capacity, and \(\Delta T\) is the temperature change.
In the exercise, we use this formula to determine how much energy it takes to heat a house from 5°C to 20°C.
Components in energy calculations include:

• Identifying the total mass of objects in the house, including air and furniture.
• Assuming a constant specific heat capacity for simplicity which permits easier calculation of total energy.
• Using the power output of a heating device (such as a heater) to calculate how quickly the temperature can be raised to the desired level.

This enables practical planning for energy needs and duration to achieve a cozy and warm home environment, allowing homeowners to optimize both time and electricity usage effectively.