(II) A heat pump is used to keep a house warm at 22°C. How much work is required of the pump to deliver 2800 J of heat into the house if the outdoor temperature is
(b) - 15°C? Assume a COP of 3.0.
COP = T_H / (T_H - T_L) .

Verified step by step guidance

1. First, understand the given values: The indoor temperature (T_H) is 22°C, which needs to be converted to Kelvin. The outdoor temperature (T_L) is -15°C, also to be converted to Kelvin. The amount of heat delivered into the house (Q_H) is 2800 J, and the Coefficient of Performance (COP) is 3.0.
2. Convert the temperatures from Celsius to Kelvin. For Kelvin, add 273.15 to the Celsius value. Thus, T_H = 22°C + 273.15 and T_L = -15°C + 273.15.
3. Recall the formula for the Coefficient of Performance (COP) of a heat pump, which is given by COP = \( \frac{T_H}{T_H - T_L} \).
4. Rearrange the COP formula to solve for the work done by the pump (W). The formula becomes W = \( \frac{Q_H}{COP} \), where Q_H is the heat delivered to the house.
5. Substitute the values of Q_H and COP into the rearranged formula to find the work required by the pump.

Key Concepts

Coefficient of Performance (COP)

The Coefficient of Performance (COP) is a measure of the efficiency of a heat pump or refrigeration system. It is defined as the ratio of the heat output (or heat removed) to the work input. A higher COP indicates a more efficient system, meaning less work is required to transfer a given amount of heat. In this case, the COP is given as 3.0, which means for every unit of work input, the heat pump delivers three units of heat.

Heat Transfer

Heat transfer refers to the movement of thermal energy from one object or substance to another due to a temperature difference. In the context of a heat pump, it involves transferring heat from the colder outdoor environment to the warmer indoor space. The efficiency of this process is influenced by the temperature difference between the heat source (outdoor) and the heat sink (indoor), which affects the work required to move the heat.

Thermodynamic Temperature Scale

The thermodynamic temperature scale is a scale that measures temperature based on the absolute zero point, where molecular motion ceases. In this problem, temperatures are given in degrees Celsius, which can be converted to Kelvin for thermodynamic calculations. The relationship between the temperatures of the heat pump's heat source and sink is crucial for calculating the COP and determining the work required to transfer heat.