# Thermodynamics - Closed Systems

## General energy equation:

\(Q - W = \Delta U + \Delta KE + \Delta PE \)

Where

Q is process dependent heat transfer

W is process dependent work done on/by system

U is the internal energy of the system

KE is the kinetic energy of the system (idk why it's KE and not \( E_k\) like how it should be)

PE is the gravitational potential energy of the system (same for this)

For stationary closed systems, change in both kinetic and potential energies are usually negligible, therefore

\(Q - W = \Delta U \)

Since internal energy is independent of process

\( \Delta U = m C_v \Delta T \)

is valid for all systems.

## Process Dependent Variables

*Isometric - Constant Volume*

\( Q = m C_v \Delta T ; W = 0\)

*Isobaric - Constant Pressure*

\( Q = m C_p \Delta T ; W = P \Delta V \)

*Isothermal - Constant Temperature*

\( Q = W = mRT ln( \frac{V_2}{V_1} ) = P_2 V_2 ln( \frac{V_2}{V_1} ) \)

*Isentropic - Constant Entropy*

\( P_1 V_1 ^\gamma = P_2 V_2 ^\gamma = Constant ; W = \frac{P_2 V_2 - P_1 V_1}{1 - \gamma} ; Q = 0\)

*Polytropic*

\( P_1 V_1 ^n = P_2 V_2 ^n = Constant ; W = \frac{P_2 V_2 - P_1 V_1}{1 - n} ; Q = \Delta U + W \)