# Heat pump work done

1. Nov 7, 2011

### issacnewton

Hi

I am trying this question from Physics gre prepbook

A heat pump is to extract from an outdoor environment at 7oC and heat the
environment indoors to 27 celcius. for each 15000 J of heat delivered indoors, the smallest
amount of work that must be supplied to the heat pump is approximately

a)500 J
b)1000 J
c)1100 J
d)2000 J
e) 2200 J

this is how I did my work.

$$T_c = (7+273) K$$

$$T_h=(27+273) K$$

the highest possible efficiency is that of carnot engine

$$\eta_{max}=1-\frac{T_c}{T_h}=\frac{1}{15}$$

let the $\eta$ be the efficiency of this pump . then

$$\eta \leqslant \frac{1}{15}$$

$$\because \; \eta=\frac{W_{eng}}{|Q_h|}$$

$$\therefore \; \frac{W_{eng}}{|Q_h|} \leqslant \frac{1}{15}$$

$$Q_h=15000 J\;\Rightarrow \; W_{eng}\leqslant \frac{15000}{15}$$

$$W_{eng} \leqslant 1000$$

which suggests that W has some maximum value but the problem is talking about the
minimum value of W. is something wrong ?