Second Law, Heat Engines and Entropy: Quick Revision

A brief note on Second Law, Heat Engines and Entropy. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Cycles. Back at the start, $\Delta U = 0$, so the net heat equals the net work, the area of the loop.
• The second law. No engine turns all its heat into work (Kelvin-Planck); heat does not flow from cold to hot by itself (Clausius).
• Reversible changes are slow, with no friction and no heat flow across a gap in temperature. Every real change is irreversible.
• Engines and refrigerators. An engine turns part of $Q_h$ into work and gives $Q_c$ to a cold sink; a refrigerator uses work to move heat from cold to hot.
• Entropy. $\Delta S = \dfrac{Q}{T}$ for heat taken in slowly. The total entropy of an isolated system never falls.

The facts at a glance

• Engine efficiency: $\eta = \dfrac{W}{Q_h} = 1 - \dfrac{Q_c}{Q_h}$
• Refrigerator: $\text{COP} = \dfrac{Q_c}{W}$, with $Q_h = Q_c + W$
• Carnot engine: $\eta = 1 - \dfrac{T_c}{T_h}$
• Best refrigerator: $\text{COP} = \dfrac{T_c}{T_h - T_c}$
• Entropy: $\dfrac{Q}{T}$; $mc\ln\dfrac{T_2}{T_1}$; $nR\ln\dfrac{V_2}{V_1}$

A worked example

A power station makes $440\ \text{MW}$ of electricity at an efficiency of $37\%$. Its steam is at $860\ \text{K}$ and it cools in a river at $305\ \text{K}$. Find the Carnot limit, the heat it uses and gives to the river each second, and the entropy it makes each second.

1. Limit. $1 - \dfrac{305\ \text{K}}{860\ \text{K}} = 0.645$, or $64.5\%$; the station reaches $37\%$.
2. Heat in. $\dfrac{440\ \text{MW}}{0.37} = 1189\ \text{MW}$. To the river: $1189\ \text{MW} - 440\ \text{MW} = 749\ \text{MW}$.
3. Entropy. The river gains $\dfrac{749\ \text{MW}}{305\ \text{K}} = 2.46\ \text{MW/K}$; the steam loses $\dfrac{1189\ \text{MW}}{860\ \text{K}} = 1.38\ \text{MW/K}$.
4. So $1.07\ \text{MW/K}$ of entropy is made each second. A Carnot engine would make none.

Practice with new numbers

A program to try

The program works out the power station of the example.

Mistakes to avoid

• Celsius in Carnot's formula. Use kelvin.
• 'The entropy of everything rises.' One body can lose entropy; only the total of an isolated system cannot fall.
• $\dfrac{Q}{T}$ for a free expansion. Use a slow path between the same states.
• COP above 1 is not 'more than 100%'. A refrigerator moves heat; it does not make energy.
• Efficiency from temperatures for a real engine. $1 - \dfrac{T_c}{T_h}$ is only the upper limit.

Check yourself

1. What is the best efficiency of an engine between $547\ ^\circ\text{C}$ and $137\ ^\circ\text{C}$?

2. Say the Clausius form of the second law in your own words.

3. $1000\ \text{J}$ of heat flows slowly into a large body at $320\ \text{K}$. By how much does its entropy rise?

4. Can you cool a kitchen by leaving the refrigerator door open?

5. Why does the entropy of the universe keep rising?