1. 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

2. 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

3. 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

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

5. 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:

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

7. 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

8. 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