Entropy at Constant Temperature

Heat flowing between two objects causes entropy change in both. Assuming the objects are big enough that the change in temperature is negligible, the change in entropy in each is $\Delta S = \frac{Q}{T}$. Their sum gives $S_{total}$.

Three cases:
• $\Delta S_{total} \lt 0$: forbidden
• $\Delta S_{total} = 0$: reversible
• $\Delta S_{total} \gt 0$: allowed but irreversible

Heat Engine

A heat engine with adjustable efficiency, $e_{actual}$. The second law forbids the actual efficiency to be higher than that of an ideal (i.e. reversable) engine because otherwise the total change in entropy is negative.

Three cases:
• $e_{actual} \lt e_{ideal}$: non-ideal
• $e_{actual} = e_{ideal}$: ideal
• $e_{actual} \gt e_{ideal}$: forbidden
where $e_{actual} = \frac{W}{|Q_{hot}|}$ and $e_{ideal} = 1- \frac{T_{cold}}{T_{hot}}$.

Carnot Engine

• The Carnot cycle consists of 2 isothermal (dark grey) and 2 adiabatic (green) processes.
• The y-axis is $\log P$ by default so that the entire Carnot cycle will fit in the limited space. You can change to the regular graph of $P$ vs $V$ if you wish.
• The cycle is set up such that the isothermal processes (step 1 and 3) change the volume by a factor of 2. In other words, by choice we make $V_b= 2V_a$ and $V_c=2 V_d$.

Things to try:
• Click on the "step 1", "step 2", ... buttons to see the Carnot cycle in action.
• Manually adjust the piston to go through the Carnot cycle yourself.
• Calculate the amount heat exchange ($|Q_{hot}|$ and $|Q_{cold}|$) with the heat bath.
• Calculate the work done by the gas ($W$) for each step of the cycle and also the total work.
• Calculate the efficiency of this engine.
• Repeat the calculations above with different $n$, $f$, and different temperatures. See how these factors affect the calculations.

The Otto Cycle

A lab manual based on this simulation is available here.

• The Otto cycle consists of 2 isochoric (constant volume, in dark grey) and 2 adiabatic (green) processes.
• Compression ratio is the ratio of the maximum to minimum volume (denoted $r$ in some conventions).
• The power of the heat source is $500W = 500J/s$.
• $Q$ is given by the power input of the heat source times $t$.

Things to try:
• Click on the "step 1", "step 2", ... buttons to see the Carnot cycle in action.
• Manually adjust the piston and heat source to go through the Carnot cycle yourself.
• Calculate the amount heat exchange with the environment in each step.
• Calculate the work done by the gas ($W$) for each step of the cycle and also the total work.
• Calculate the efficiency of this engine.
• Repeat the calculations above with different $n$, $f$, and different temperatures. See how these factors affect the calculations.

The Rectangular Cycle

A lab manual based on this simulation is available here.

• The Rectangular cycle consists of 2 isochoric (constant volume) and 2 isobaric processes.
• Compression ratio is the ratio of the minimum to maximum volume.
• A background of isothermal curves are drawn in grey at $200K$ intervals.
• The power of the heat source is $500W = 500J/s$.
• $Q$ is given by the power input of the heat source times $t$.

Things to try:
• Click on the "step 1", "step 2", ... buttons to see the Carnot cycle in action.
• Manually adjust the piston and heat source to go through the Carnot cycle yourself.
• Calculate the amount heat exchange with the environment in each step.
• Calculate the work done by the gas ($W$) for each step of the cycle and also the total work.
• Calculate the efficiency of this engine.
• Repeat the calculations above with different $n$, $f$, and different temperatures. See how these factors affect the calculations.