Final answer:
To operate at the maximum thermodynamic efficiency and supply 1,000 W of heat, the electrically powered heat pump must consume 100 W of electrical power, as it has a coefficient of performance of 10 given the temperature difference between the heat source and the room.
Step-by-step explanation:
To determine the power consumption of the heat pump operating between 270 K and 300 K with maximum thermodynamic efficiency, we utilize the concept of the coefficient of performance (COP) for a heat pump. The COP is defined as the amount of heat transferred to the hot reservoir (Qh), divided by the work input (W).
The COP for a heat pump is given by the formula:
COPhp = Qh/W = Th/(Th - Tc)
Where:
- Th is the temperature of the hot reservoir in kelvins (300 K in this case),
- Tc is the temperature of the cold reservoir in kelvins (270 K in this case).
Using these values, we can calculate the COP for this heat pump:
COPhp = 300/(300 - 270) = 300/30 = 10
To find out how much electrical power must be consumed by the heat pump to supply 1,000 W of heat to the room, we can rearrange the COP formula to solve for W:
W = Qh/COPhp = 1,000 W / 10 = 100 W
Therefore, the heat pump must consume 100 W of electrical power to supply 1,000 W of heat, assuming maximum theoretical efficiency.