Rate Equation

First Law of Thermodynamics:

• Differential form: $dE=d(KE)+d(PE)+dU=\delta Q-\delta W$
• Integrated between two states: $\mathrm{\Delta }E=\mathrm{\Delta }(KE)+\mathrm{\Delta }(PE)+\mathrm{\Delta }U=\int \delta Q-\int \delta W$

Derivation:

• Normalize by time increment
• Limit when $\delta t\to 0$

Rate Equation:

Heat transfer rate ($\dot{Q}$):

Rate of work(i.e. power) ($\dot{W}$):

Second Law of Thermodynamics:

• Differential form: $dS\ge \frac{\delta Q}{T}$
• Integrated form: $\mathrm{\Delta }S\ge \int \frac{\delta Q}{T}$

Continuous Process:

• Single heat transfer: $\frac{dS}{dt}\ge \frac{\dot{Q}}{T}$
• Multiple heat transfers: $\frac{dS}{dt}\ge \sum \frac{\dot{Q}}{T}$

Steady State: