what is the work done, heat added, and change in internal energy?

Question

In a polytropic process, the volume of a gas is changed in such a way that pvⁿ = constant for some
value of n. An isothermal process can be considered a polytropic process for n=1. In a polytropic
process, what is the work done, heat added, and change in internal energy? Your answer may include
the initial condition (P₀ V₀ T₀)  final condition (P_f  V_f T_f)  the compression ratio (R= V_v / V_f) constants for the gas (K_B C_V  C_P U), n and other relevant physical constants.

Answer 1

Work done by gas:

 

W = ∫P(V)dV, from V₀ to V_f

 

PVⁿ = constant = P₀V₀ⁿ ⇒

 

P(V) = P₀V₀ⁿ/Vⁿ

 

Plug this into the integral and evaluate with limits V₀ and V_f.

 

Change in internal energy:

 

ΔU = ∫C_v(T)dt from T₀ to T_f

 

Heat transfer into gas:

 

Q = ΔU + W