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 quasistatically, 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 equation (1),
we get
`therefore 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_(r))`
`=(1)/(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)=(1)/(1-gamma)[P_(f)V_(f)-P_(i)V_(i)]" "...(2)`
From ideal gas law,
`P_(f)V_(f)=mu RT_(f) and P_(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 expansion. 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.