Colby College. Application of the First Law to Ideal Gases. Calculate q,w, ∆U, and
∆H for ideal gas processes: dU = ñq + ñw. dU = Cv dT dH = Cp dT. ∆U = q + w.
Application of the First Law to Ideal Gases Calculate q,w, ∆U, and ∆H for ideal gas processes: dU = ñq + ñw dU = Cv dT dH = Cp dT ∂U ∆U = q + w for any process since = 0 ∂VT Isothermal Reversible Expansion dT = 0 so dU = and dH = q = –w V2 V1
w = – nRT ln P2V2 = P1 V1
V2 P1 V1 = P2
or
P1 w = – nRT lnP
2
Adiabatic Reversible Expansion ñq = so dU = ñw = – P dV dU = – P dV or dU = Cv dT Cv dT = – P dV dT dV Cv T = – nR V dT dV ⌠T 2 ⌠V = – 2 nR Cv T V ⌡T 1 ⌡ V1 V2
T2
Cv ln T
|T
= – nR ln V 1
|V
1
T2 V2 Cv ln = – nR ln T1 V1
Cv T2 V2 Method 1: nR lnT = – lnV c = Cv/nR 1 1 Cv/nR Cv/nR T2Cv/nR V1 or = V = V 2T2 1T1 T1 V2
Colby College
First Law and Ideal Gases
Method 2: Reversible adiabatic: P = Pext dH = dU + PdV + VdP = ñq – P dV + PdV + VdP = V dP dH = CpdT = V dP
dP CpdT = nRT P
T2 P2 Cp ln T = nR lnP
Cp T2 P2 = ln ln nR T1 P1
1
1
T2Cp/nR P2 = T1 P1
Relate cst V and cst P processes: Cp ln(T2/T1) nR ln (P2/P1) γ = Cp/Cv = Cv ln(T2/T1) – nR ln(V2/V1) γ=