**First law for an engine cycle says:**

W_{out} = Q_{h} – Q_{c}

,where

W_{out} is work output by engine

Q_{h} is heat input from hot reservoir

Q_{c} is heat output to cold reservoir

**Thermal efficiency is defined to be the ratio of work output to the heat input from the hot reservoir per cycle,**

$$\begin{aligned} e \, &= \frac{W_{\text{out}}}{Q_{h}} \\ &= \frac{Q_{h} \, – Q_{c}}{Q_{h}} \\ &= 1 \, – \frac{Q_{c}}{Q_{h}} \end{aligned}$$

This thermal efficiency is simply 1 minus the fraction of heat rejected into the cold reservoir.

### Heat Pump: Coefficients of performance

**First law for a heat pump is:**

$$W_{\text{in}} = Q_{h} – Q_{c}$$

Here, Q_{h} is defined as heat output to hot reservoir and Q_{c} is heat input from cold reservoir, because we are dealing with a heat pump.

In the **heating mode**, where we are interested in the ratio of heat output into the hot reservoir to the work input per cycle, we define coefficient of performance (for heating):

$$K_{h} = \frac{Q_{h}}{W_{\text{in}}} = \frac{Q_{h}}{Q_{h} – Q_{c}}$$

In the **cooling mode**, where we are interested in the ratio of heat extracted from the cold reservoir to the work input per cycle, we define coefficient of performance (for cooling):

$$K_{c} = \frac{Q_{c}}{W_{\text{in}}} = \frac{Q_{c}}{Q_{h} – Q_{c}}$$