### Work Done in Basic Thermodynamic Processes

Basic goal:  determine dE, dQ, and dW for general thermodynamics processes. We now study 3 fundamental processes.

#### Constant Volume (Isochoric)

A constant volume process is the vertical path dV = 0 in the P-V plane---up if heat is added and down if heat is removed. Because dV = 0, the work done is   dW = - P dV = 0

The First Law of Thermodynamics then states:   dE= dQ + dW = dQ.

For a monatomic ideal gas:  E =
 3 2
NkT   →   dQ = dE =
 3 2
Nk dT

#### Constant Pressure (Isobaric)

A constant pressure process is a horizontal path in the P-V diagram---right for expansion and left for compression.   Example: a gas in a container sealed with a freely-sliding massive piston.

The work done during gas expansion is:   dW = - P dV = - Nk dT

The First Law gives: dQ = dE - dW =
 3 2
Nk dT + Nk dT =
 5 2
Nk dT

#### Constant Temperature (Isothermal)

A constant temperature process is an isothermal path in the P-V diagram---a hyperbolic isotherm.   Example: a gas in a container that is immersed in a constant-temperature bath is allowed to expand slowly, or is compressed slowly.

At constant temperature, the pressure of an ideal gas is:   P = NkT/V.

The work done on the gas is:
W = - vivf P dV = - NkT vivf
 1 V
dV
W = - NkT ln (
 Vf Vi
) = - Pi Vi ln (
 Vf Vi
) = - Pf Vf ln (
 Vf Vi
)

The First Law gives:   dE = dQ + dW = 0   →   dQ = - dW.

In an isothermal process, there is no change in the internal energy of an ideal gas.

Important: The work done during any thermodynamic process is path dependent (see transparency).