On a cold day, a heat pump absorbs heat from the outside air at 14°F (−10°C) and transfers it into a home at a temperature of 86°F (30°C). Determine the maximum κ of the heat pump.

To determine the maximum coefficient of performance (κ) for a heat pump, we can use the Carnot efficiency equation for a heat pump, which is a theoretical maximum efficiency that any heat pump can achieve. The equation is as follows:

κ_max = T_warm / (T_warm - T_cold)

Here, κ_max is the maximum coefficient of performance, T_warm is the absolute temperature of the warm reservoir (inside the home) in Kelvins, and T_cold is the absolute temperature of the cold reservoir (outside air) in Kelvins. It's important to note that temperatures should be converted to Kelvins for this calculation.

First, convert the given temperatures from degrees Fahrenheit and Celsius to Kelvin: T_warm (in Fahrenheit) = 86°F -> T_warm (in Kelvin) = (86°F − 32) × 5/9 + 273.15 ≈ 303.15 K T_cold (in Celsius) = −10°C -> T_cold (in Kelvin) = −10°C + 273.15 = 263.15 K

Now, we can plug these values into the formula: κ_max = T_warm / (T_warm - T_cold) κ_max = 303.15 K / (303.15 K - 263.15 K) κ_max = 303.15 K / 40 K κ_max ≈ 7.58

So, the maximum coefficient of performance (κ) for a heat pump operating between these temperatures is approximately 7.58.