Polytropic Processes A polytropic process takes the general form of the common thermodynamic processes, and is defined by the equation pV n = C or p1V1n = p2V2n where n is an index we may manipulate to get our common thermodynamic processes. Source: nptel.ac.in Note: as n approaches ∞, p has less and less influence; when n = ∞, p has no influence and we thus have an isobaric process. Other n values (x in the graph) are possible, but not common. For a physical interpretation of these other values, check the Wikipedia page on polytropic processes. Because polytropic processes are general representations of thermodynamic processes, any quantities we calculate for a polytropic process will be a general equation for that quantity in all other thermodynamic processes. We proceed to find the work and heat associated with a polytropic process. We start with the definition of work: Z V2 w=− p dV V1 The polytropic equation pV n = C allows us to manipulate our formula for work in the following manner, as long as n 6= 1 Z V2 w=− V1 C dV = −C Vn Z V2 V −n dV = − V1 C 1−n V2 C 1−n V2 = V V V V1 1 1−n n−1 Evaluating this expression, w= CV21−n − CV11−n C V21−n − V11−n = n−1 n−1 We let the first C be p2 V2n and the second C be p1 V1n , such that the exponents cancel to obtain w= p2 V2 − p1 V1 R(T2 − T1 ) p2 V2n V21−n − p1 V1n V11−n = = n−1 n−1 n−1 where the last equality follows from the ideal gas law pV = RT and our derivation for the work of a polytropic process is complete. Next, in order to find the heat of a polytropic process, we first find ∆U and use the first law to find q. Knowing that CV = R γ−1 we can write ∆U = CV ∆T = R(T2 − T1 ) γ−1 Then we have q = ∆U − w = R(T2 − T1 ) R(T2 − T1 ) − = R(T2 − T1 ) γ−1 n−1 1 1 − γ−1 n−1 We can simplify this further by rewriting our final term as q= R(n − γ) (T2 − T1 ) (γ − 1)(n − 1) and our derivation for the heat of a polytropic process is complete. This form for the heat of a polytropic process is also suggestive of the heat capacity of a polytropic process, for which q = C∆T . We can thus see by inspection that the polytropic heat capacity Cn is given by Cn = R(γ − n) (n − γ) = CV (n − 1)(γ − 1) (n − 1) and thus we can write the formula for the heat of a polytropic process as q = CV (n − γ) (T2 − T1 ) = Cn (T2 − T1 ) (n − 1) We can justify our formulae by plugging in values of n and seeing if the formulae give the expected results. In particular, n q w isobar 0 CP (T2 − T1 ) −R(T2 − T1 ) isotherm 1 −RT ln V2 /V1 * RT ln V2 /V1 * adiabat γ 0 CV (T2 − T1 ) isochor ∞ CV (T2 − T1 ) 0