Maximum Work

The Kelvin-Planck Statement of the Second Law

${\eta }_{th}=\frac{W^{\prime }}{Q_H^{\prime }}=\frac{Q_H^{\prime }-Q_L^{\prime }}{Q_H^{\prime }}=1-\frac{Q_L^{\prime }}{Q_H^{\prime }}$

Maximum Thermal Efficiency

${\eta }_{\text{th,carnot}}=\left(1-\frac{T_L}{T_H}\right)$
$w_{max}^{\prime }={\eta }_{\text{th,carnot}}{Q}_H^{\prime }$
As $T_H$ → $\infty$ $Q^{\prime }$ is as useful as $W^{\prime }$
$T_H\to T_L$ $Q^{\prime }$ is useless.

Attainable Efficiency

A 75% Carnot efficiency is considered attainable

Refrigerators and Heat Pumps

Both refrigerators and heat pumps move energy fro a cold reservoir to a hotter one using work. The only difference here is the goal (move heat from the cold space, or get heat into the hot space)

Equations
Refrigerators $CO{P}_R=\frac{Q_L^{\prime }}{W^{\prime }}=\frac{Q_L^{\prime }}{Q_H^{\prime }-Q_L^{\prime }}$
$CO{P}_{R,carnot}=\frac{1}{\frac{T_H}{T_L}-1_{}}$
Heat Pumps
$CO{P}_{HP}=\frac{Q_H^{\prime }}{Q_H^{\prime }-Q_L^{\prime }}$
$CO{P}_{HP,carnot}=\frac{1}{1-\frac{T_L}{T_H}}$