Give the equation of state for an adiabatic process.

Explanation

Work done in an adiabatic process: Consider `mu` moles of an ideal gas enclosed in a cylinder having perfectly non conducting walls and base. A frictionless and insulating piston of cross sectional area A is fitted in the cylinder.
Let W be the work done when the system goes from the initial state `(P_(i)V_(i)T_(i))` to the final state `(P_(f)V_(f)T_(f))` adiabatically.
`W=int_(V_(i))^(V_(f))PdV" ...(1)"`

By assuming that the adiabatic process occurs quasi - statically, at every stage the ideal gas law is valid. Under this condition, the adiabatic equation of state is `PV^(gamma)="constant (or) "P=("constant")/(V^(gamma))` can be substituted in the equation (1), we get
`W_("adia")=int_(V_(i))^(V_(f))("constant")/(V^(gamma))dV="constant "int_(V_(i))^(V_(f))V^(-gamma)dV`
`="constant "[(V^(-gamma+1))/(-gamma+1)]_(V_(i))^(V_(f))=("constant")/(1-gamma)[(1)/(V_(f)^(gamma-1))-(1)/(V_(i)^(gamma-1))]`
`=(1)/(1-gamma)[("constant")/(V_(f)^(gamma-1))-("constant")/(V_(i)^(gamma-1))]`
`"but, "P_(i)V_(i)^(gamma)=P_(f)V_(f)^(gamma)="constant"`
`therefore" "W_("adia")=(1)/(1-gamma)[(P_(f)V_(f)^(gamma))/(V_(f)^(gamma-1))-(P_(i)V_(i)^(gamma))/(V_(i)^(gamma-1))]`
`W_("adia")=(1)/(1-gamma)[P_(f)V_(f)-P_(i)V_(i)]" ...(2)"`
From ideal gas law, `P_(f)V_(f)=muRT_(f) andP_(i)V_(i)=muRT_(i)`
Substituting in equation (2), we get
`therefore" "W_("adia")=(muR)/(gamma-1)[T_(i)-T_(f)]`
In adiabatic expansion, work is done by the gas. i.e., `W_("adia")` is positive. As `T_(i) gt T_(f)`, the gas cools during adiabatic compression.
In adiabatic compression, work is done on the gas. i.e., `W_("adia")` is negative. As `T_(i) lt T_(f)`, the temperature of the gas increases during adiabatic compression.

To differentiate between isothermal and adiabatic curves in PV diagram, the adiabatic curve is drawn along with isothermal curve for `T_(f)` and `T_(i)`. Note that adiabatic curve is steeper than isothermal curve. This is because `gamma gt 1` always.