dU = dq + dw Now 1st Law becomes: dU = CvdT + PdV Can also define constant – pressure heat capacity: CP ≡ (∂q/∂T)P Finally: CP – CV = R

If system can change volume, At constant volume: dU = dq If system can change volume, dU ≠ dq Some heat into the system is converted to work ∴ dU < dq Constant pressure processes much more common than constant volume processes

dU = dq + dw = dq + PdV Enthalpy, H defined as: H ≡ U + PV At constant pressure: dH = dqP For a measurable change: ΔH = qP Properties of H: absolute H cannot be determined state function H is an extensive physical property

function of temperature Plot of enthalpy as a function of temperature H = U + PV Slope = (∂H/∂T)P The heat capacity at constant pressure:

Variation of Enthalpy with Temperature At constant pressure: dH = CP dT ∫dH = CP ∫dT (accurate only for a monatomic gas and/or small changes in T) ΔH = CP ΔT For molecular gases and large changes in T:

Use an approximate empirical expression: CP = a + bT + cT-2 e.g. For N2: a = 1021, b = 134.6, c = -17.9 (all in J K-1 kg-1) Determine ΔH when air is heated from -25°C to 20°C. (Assume air is pure N2.)

Consider dry air in a layer heating and expanding: The air expands and rises doing work vs gravity U increases according to dU = CvdT Work is done according to w = PdV Assume air is N2 and O2 alone, then heat (dq) added is partitioned in a ratio of 5:2 as: dq = dU + PdV U receives 2.5R units of heat, w receives R units

Consider dry air in a layer heating and expanding: A more general expression for a moving dry air parcel which varies in P as it rises or sinks: dΦ = - V dP dq = dH - V dP dH = Cp dT dq = d(H + Φ) = d(Cp dT + Φ) d(H + Φ) ≡ dry static energy