Question

Derive the work done in an adiabatic process.

Answer 1

Work done in an adiabatic process: Consider µ 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.

        Work done in an adiabatic process

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^γ = constant (or)

P = `"constant"/"V"^γ` can be substituted in the equation (1), we get

∴ W_adia = `int_("V"_"i")^("V"_"f") "constant"/"V"^γ "dV" = "constant" int_("V"_"i")^("V"_"f") "V"^-γ "dV"`

= `"constant" [("V"^(-γ + 1))/(-γ + 1)]_("V"_"i")^("V"_"f") = "constant"/(1 - γ) [1/("V"_"f"^(γ - 1)) - 1/("V"_"i"^(γ - 1))]`

= `1/(1 - γ) ["constant"/("V"_"f"^(γ - 1)) - "constant"/("V"_"i"^(γ - 1))]`

But, `"P"_"i""V"_"i"^γ = "P"_"f""V"_"f"^γ = "constant"`

∴ W_adia = `1/(1 - γ) [("P"_"f""V"_"f"^γ)/("V"_"f"^(γ - 1)) - ("P"_"i""V"_"i"^γ)/("V"_"i"^(γ - 1))]`

W_adia = `1/(1 - γ) ["P"_"f""V"_"f" - "P"_"i""V"_"i"]` .......(2)

From ideal gas law, P_fV_f = μRT_f and P_iV_i = μRT_i

Substituting in equation (2), we get

∴ W_adia = `(μ"R")/(γ - 1) ["T"_"i" - "T"_"f"]` ..........(3)

In adiabatic expansion, work is done by the gas. i.e., W_adia is positive. As T_i > 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 < T_f the temperature of the gas increases during adiabatic compression.

PV diagram - Work done in the adiabatic process

To differentiate between isothermal and adiabatic curves in the PV diagram, the adiabatic curve is drawn along with the isothermal curve for T_f and T_i. Note that the adiabatic curve is steeper than an isothermal curve. This is because γ > 1 always.