Polytropic Processes

a.cyclohexane.molecule

November 2015

A polytropic process takes the general form of the common thermodynamic processes, and is defined by the equation n n n pV = C or p1V1 = p2V2 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 pro- cesses, 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

V V Z 2 C Z 2 C V C V w = − dV = −C V −n dV = − V 1−n 2 = V 1−n 2 n V1 V1 V1 V V1 1 − n n − 1 Evaluating this expression,

C CV 1−n − CV 1−n w = V 1−n − V 1−n
= 2 1 n − 1 2 1 n − 1

n n We let the first C be p2V2 and the second C be p1V1 , such that the exponents cancel to obtain p V nV 1−n − p V nV 1−n p V − p V R(T − T ) w = 2 2 2 1 1 1 = 2 2 1 1 = 2 1 n − 1 n − 1 n − 1 where the last equality follows from the ideal gas law pV = RT and our deriva- tion 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 R C = V γ − 1 we can write R(T − T ) ∆U = C ∆T = 2 1 V γ − 1

Then we have R(T − T ) R(T − T )  1 1  q = ∆U − w = 2 1 − 2 1 = R(T − T ) − γ − 1 n − 1 2 1 γ − 1 n − 1

We can simplify this further by rewriting our final term as

R(n − γ) q = (T − T ) (γ − 1)(n − 1) 2 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 R(γ − n) (n − γ) C = = C n (n − 1)(γ − 1) V (n − 1) and thus we can write the formula for the heat of a polytropic process as

(n − γ) q = C (T − T ) = C (T − T ) V (n − 1) 2 1 n 2 1

We can justify our formulae by plugging in values of n and seeing if the formulae give the expected results. In particular,

isobar isotherm adiabat isochor n 0 1 γ ∞

q CP (T2 − T1) −RT ln V2/V1* 0 CV (T2 − T1)

w −R(T2 − T1) RT ln V2/V1* CV (T2 − T1) 0

* Not obtained from our present derivation. Questions: 1. Why are our polytropic process formulae invalid for n = 1? 2. Why is there no benefit to calculating changes in state functions using poly- tropic processes?