Consider a house with a floor space of \(200 \mathrm{~m}^{2}\) and an average height of \(3 \mathrm{~m}\) at sea level, where the standard atmospheric pressure is \(101.3 \mathrm{kPa}\). Initially the house is at a uniform temperature of \(10^{\circ} \mathrm{C}\). Now the electric heater is turned on, and the heater runs until the air temperature in the house rises to an average value of \(22^{\circ} \mathrm{C}\). Determine how much heat is absorbed by the air assuming some air escapes through the cracks as the heated air in the house expands at constant pressure. Also, determine the cost of this heat if the unit cost of electricity in that area is $$\$ 0.075 / \mathrm{kWh}$$.

First, we need to calculate the initial volume of the air inside the house. The volume (V) of any space can be found by multiplying the area (A) by the height (h). In this case: V = A * H Where A = 200 m² and H = 3 m. V = 200 m² * 3 m = 600 m³

To find the final volume of air, we can use the formula for the ideal gas law, which is: PV = nRT We know that the initial and final pressures (P) are constant at 101.3 kPa, and the temperature changes from 10°C to 22°C. The universal gas constant (R) is 8.314 J/(mol*K). We want to find the final volume (V₂) and the number of moles (n₂) of air. To do this, we will first convert the initial and final temperatures to Kelvin by adding 273.15: T₁ = 10°C + 273.15 = 283.15 K T₂ = 22°C + 273.15 = 295.15 K From the ideal gas law, we can calculate n₁V₁/T₁ and n₂V₂/T₂: n₁V₁/T₁ = n₂V₂/T₂ Since the pressures are equal, we can substitute the initial volume (V₁) into the formula: n₂V₂ = n₁V₁T₂/T₁ Since we are only interested in finding the final volume (V₂), V₂ = V₁T₂/T₁ Substitute the values: V₂ = 600 m³ * (295.15 K / 283.15 K) = 626.23 m³

Now, we will determine the heat (Q) absorbed by the air using the formula: Q = n*C_p*ΔT Where C_p is the specific heat capacity of air at constant pressure (1.005 kJ/kg*K), n is the number of moles (n = n₂ - n₁), and ΔT is the change in temperature (ΔT = T₂ - T₁). We will first find n by dividing the difference in volumes (V₂ - V₁) by the molar volume of air at standard conditions (22.4 L/mol = 0.0224 m³/mol): n = (V₂ - V₁) / 0.0224 m³/mol = (626.23 m³ - 600 m³) / 0.0224 m³/mol ≈ 1169.64 mol Now we can find Q: Q = n * C_p * ΔT = 1169.64 mol * 1.005 kJ/mol*K * (295.15 K - 283.15 K) ≈ 16447.06 kJ

Finally, we can calculate the cost of the absorbed heat. First, we need to convert the heat from kJ to kWh: 1 kWh = 3600 kJ 16447.06 kJ * (1 kWh / 3600 kJ) ≈ 4.57 kWh Now we can calculate the cost using the given unit cost of electricity ($0.075/kWh): Cost = 4.57 kWh * \(0.075/kWh ≈ \)0.34 Thus, the cost of heating the air in the house from 10°C to 22°C is approximately $0.34.