Transcription of T V v /nR
1 Application of the First Law to Ideal Gases Calculate q,w, U, and H for ideal gas processes: dU = q + w dU = Cv dT dH = Cp dT. U . U = q + w for any process since = 0. V T. Isothermal Reversible Expansion dT = 0 so dU = and dH =. q = w V2 . w = nRT ln . V1 . V2 P1. P2V2 = P1 V1 or V1 = P2. P1 . w = nRT ln P . 2 . Adiabatic Reversible Expansion q = so dU = w = P dV. dU = P dV or dU = Cv dT. Cv dT = P dV. dT dV. Cv T = nR V. T 2 dT V dV. Cv = 2 nR. T 1 T V1 V. T2 V2. Cv ln T |T 1. = nR ln V |V. 1. T2 V2 . Cv ln = nR ln . T1 V1 . Cv T2 V2.
2 Method 1: nR ln T = ln V c = Cv/nR. 1 1 . T2 Cv/nR V1 Cv/nR Cv/nR. = or V 2T2 = V1T1. T1 V2 . colby college First Law and Ideal Gases Method 2: Reversible adiabatic: P = Pext dH = dU + PdV + VdP = q P dV + PdV + VdP = V dP. dP. dH = CpdT = V dP CpdT = nRT P. T2 P2 Cp T2 P2 . Cp ln T = nR ln P ln = ln . 1 1 nR T1 P1 . T2 Cp/nR P2 . = . T1 P1 . Relate cst V and cst P processes: Cp ln(T2/T1) nR ln (P2/P1). = Cp/Cv =. Cv ln(T2/T1) nR ln(V2/V1). ln (P2/P1). =. ln(V1/V2). ln(V1/V2) = ln(P2/P1). V1 P2 . = . V2 P1 .. P2V2 = P1V 1 P1. 8. P (bar).
3 6. isothermal 4 P2V2 = P1V1. 2 adiabatic . P2V2 = P1V1. 0 V1 2 4 V2 6. V (L). Adiabatic Irreversible Expansion: Example: Pext = constant = P2. U = w U = Cv (T2 T1). U = Pext (V2 V1) Cv T = Pext V. Cv(T2 T1) = Pext(V2 V1). nRT2 nRT1 . Cv(T2 T1) = P2 P P . 2 1 . 3 5. monatomic gas: Cv= nR diatomic gas (neglecting vibration): Cv= nR. 2 2. colby Colleg
