Structural Biochemistry/Equation for Process Calculations for Ideal Gases

Equation for Process Calculations for Ideal Gases
For reversible, closed-system, work is given by

dW=-PdV

For ideal gases, the first law can be written by

dQ+dW=C_v dT

From two equations above, we get

dQ=C_v dT+PdV

These three equations can be applied to four types of processes: isothermal, isobaric, isochoric and adiabatic.

Isothermal Process


Isothermal process deals with closed-system that has constant temperature. So ΔT=0:

ΔU=ΔH=0

Q=RTln V_2/V_1 =-RTln P_2/P_1

W=-RTln V_2/V_1 =RTln P_2/P_1

Q=-W        (constant T)

Therefore,

Q=-W=RT ln V_2/V_1 = -RT ln P_2/P_1  (constant T)

Isobaric Process


Isobaric process deals with closed-system that has constant pressure. So ΔP=0.

ΔU=∫▒〖C_v dT〗    and     ΔH=∫▒〖C_p dT〗

Q=∫▒〖C_p dT〗   and    W=-R(T_2-T_1)

Therefore,

Q=ΔH=∫▒〖C_p dT〗      (constant P)

Isochoric Process


Isochoric process deals with closed-system that has constant volume. So ΔV=0.

ΔU=∫▒〖C_v dT〗    and     ΔH=∫▒〖C_p dT〗

Q=∫▒〖C_v dT〗   and    W=-∫▒PdV=0

Therefore,

Q=ΔU=∫▒〖C_v dT〗      (constant V)

Adiabatic Process
Adiabatic process deals with closed-system that has no heat transfer between the system and the surroundings. So ΔQ=0.

dT/T= -R/C_v   dV/V

T_2/T_1 =(V_1/V_2 )^(R⁄C_v )

T_2/T_1 =(P_2/P_1 )^(R⁄C_p ) and   P_2/P_1 =(V_1/V_2 )^(C_p⁄C_v )

The following equations apply to ideal gases with constant heat capacities that undergo mechanically reversible adiabatic expansion or compression.

〖TV〗^(γ-1)=constant      〖TP〗^(((1-γ))⁄γ)=constant       〖PV〗^γ=constant

γ≡ C_p/C_v

For any adiabatic closed-system,

dW=dU= C_v dT

W= △U= C_v△T

γ ≡ C_p/C_v =  (C_v+R)/C_v =1+  R/C_v       or     C_v=  R/(γ-1)

W= C_v△T= (R△T)/(γ-1)

W= (〖RT〗_2-〖RT〗_1)/(γ-1)=  (P_2 V_2-P_1 V_1)/(γ-1)

For mechanically reversible process,

W= (P_1 V_1)/(γ-1) [(P_2/P_1 )^(γ-1)-1]=  (RT_1)/(γ-1) [(P_2/P_1 )^(((γ-1))⁄γ)-1]

Diabatic Process
Opposite of adiabatic process There is heat transfer

Polytropic Process
Polytropic process deals with a model of some versatility. So δ=constant.

〖PV〗^δ=constant

〖TV〗^(δ-1)=constant           〖TP〗^(((1-δ))⁄δ)=constant

W= (RT_1)/(δ-1) [(P_2/P_1 )^(((δ-1))⁄δ)-1]

Q= ((δ-γ)RT_1)/((δ-1)(γ-1))  [(P_2/P_1 )^(((δ-1))⁄δ)-1]

Reference
Smith, J. M., and Ness H. C. Van. Introduction to Chemical Engineering Thermodynamics. New York: McGraw-Hill, 1987. Print.