#### \(pV\) diagrams

To avoid needlessly long descriptions, we can use diagrams to show the processes that take place in a container of gas.

A graph of pressure vs volume enables us to show all possible types of process because of the interrelationship between pressure, volume and temperature in the ideal gas law: \(pV\propto T\)

We can use the first law to consider the changing heat, internal energy and work done: \(Q=\Delta U+p\Delta V\)

#### Isovolumetric

An *isovolumetric *process takes place at constant volume. This is represented by a vertical line (isochore) on a \(pV\) diagram.

In this example, \(\textrm{B}\to\textrm{C}\) represents an increase in pressure and temperature.

\(p\propto T\)

- \(p\) is pressure (Pa)
- \(T\) is temperature (K)

\({p_1\over T_1}={p_2\over T_2}\)

Since internal energy is increasing, heat must be gained by the gas. No work is done.

#### Isobaric

An *isobaric *process takes place at constant pressure. This is represented by a horizontal line (isobar) on a \(pV\)diagram.

In this example, \(\textrm{A}\to\textrm{B}\) represents an expansion and increase in temperature.

\(V\propto T\)

- \(V\) is volume (m
^{3}) - \(T\) is temperature (K)

\({V_1\over T_1}={V_2\over T_2}\)

Since internal energy is increasing and work is done on the surroundings, heat must be gained by the gas.

#### Isothermal

An *isothermal *process takes place at constant temperature. This is represented by a line of inverse proportion (isotherm) on a \(pV\) diagram.

In this example, \(\textrm{E}\to\textrm{F}\) represents an expansion with a reduction in pressure.

\(p\propto {1\over V}\)

- \(p\) is pressure (Pa)
- \(V\) is volume (m
^{3})

\(p_1V_1=p_2V_2\)

Since internal energy is constant and work is done on the surroundings, heat must be gained by the gas. Isothermal processes are conducted slowly, to avoid increasing the kinetic energy of the particles.

#### Adiabatic

No heat is gained or lost during an *adiabatic* process. This is represented by an adiabat on a \(pV\) diagram, which has a steeper gradient throughout than an isotherm.

In this example, \(\textrm{G}\to\textrm{H}\) represents an expansion with a reduction in pressure and temperature.

The following equation can be derived for monatomic gases:

\(p\propto {1\over V^{5\over 3}}\)

- \(p\) is pressure (Pa)
- \(V\) is volume (m
^{3})

\(p_1V_1^{5\over 3}=p_2V_2^{5\over 3}\)

Since no heat enters or leaves, the decrease in internal energy must be equal to the work done on the surroundings. Adiabatic processes are conducted rapidly, to prevent sufficient time for heat to enter or leave.