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.

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 \(10,000 \mathrm{~W}\))

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

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.

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).