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 thumb|1px|From left to right the lines signify: isochoric, adiabatic, isothermal, and isobaric. 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 1px|thumbnail|From left to right the lines signify: isochoric, adiabatic, isothermal, and isobaric. 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 1px|thumbnail|From left to right the lines signify: isochoric, adiabatic, isothermal, and isobaric. 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 1px|thumbnail|From left to right the lines signify: isochoric, adiabatic, isothermal, and isobaric. 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.