Adiabatic process: This is a process in which no heat flows into or out of the system (Q = 0). But the gas can expand by spending its internal energy or gas can be compressed through some external work. So the pressure, volume and temperature of the system may change in an adiabatic process. · For an adiabatic process, the first law becomes `Delta U` = W. This implies that the work is done by. the gas at the expense of internal energy or work is done on the system which increases its internal energy
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 `P_V` , = Constant (or)
`P = ("constant ")/(Vr)` 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_fV_f^gamma =` constant
` therefore W_(adia) = (1)/(1 - gamma) [ (P_fV_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 ] ` ...(1)
From ideal gas law `P_f V_f = mu RT_f " and " P_i V_i = mu RT_i `
substituting in eqn (2) , we get
`W_(adia ) = (mu R)/(gamma -1) [ T_i - T_f]` ....(3)
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 .
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` > 1 always.
