H.2 Euler-Maclaur in Sum Formula
8.2 Sketches of the Thermodynamic Functions
We can gain more insight into monocomponent systems by sketching the thermodynamic functionsμ,h, andsas functions ofpandT. For a phase diagram of the form ofFigure 8–1, we choose three constant pressures,p1,p2, andp3 as indicated inFigure 8–4, and then discussμ,h, andsas a functionTat each of these pressures.
Along a line of constantp,μis a continuous function ofT. According to Eq. (8.7), its slope is−s. Butsis discontinuous at a coexistence curve, soμhas a discontinuity of slope asT crosses a coexistence curve.Figure 8–5shows a sketch ofμas a function ofT along the linep1inFigure 8–4.
To quantify the behavior ofsandh, we must view them as functions ofTandp. Within a bulk phase,
ds= ∂s
∂T
p
dT+ ∂s
∂p
T
dp= cp
T dT+ ∂s
∂p
T
dp, (8.24)
wherecpis the heat capacity per mole at constant pressure. From Eq. (8.7) we have the Maxwell relation
∂s
∂p
T= − ∂v
∂T
p= −vα, (8.25)
p
T Tt, pt
Tc, pc
S L
V p1
p2 p3
FIGURE 8–4 Constant pressure pathsp1,p2, andp3on a phase diagram for the monocomponent system ofFigure 8–1. The chemical potentialμis continuous along these paths, but its slope,−s, changes asTcrosses a coexistence curve.
116 THERMAL PHYSICS
à
TSV T S
V
FIGURE 8–5 Sketch of the chemical potentialμas a function ofT along the linep= p1inFigure 8–4. The full line corresponds to the stable solid and vapor phases. The dashed lines are extrapolations into unstable regions of superheated solid and supercooled vapor, intended to emphasize the discontinuity of slope of the full line at the phase transition. The stable phase is solid (S) forT≤TSVand vapor (V) forT≥TSV.
whereαis the coefficient of expansion. Thus ds=cp
T dT−vαdp. (8.26)
For the enthalpy per mole, we have dh=Tds+vdp=T
∂s
∂T
p
dT+
T ∂s
∂p
T
+v
dp. (8.27)
Thus
dh=cpdT+v(1−Tα)dp. (8.28)
For the sake of consistency of Eqs. (8.26) and (8.28), note that substitution into dμ=dh− sdT−Tdsleads back to Eq. (8.6). Thus within a single phase at constant pressure we have
h(T2)−h(T1)= T2 T1
cpdT≈cp(T2−T1); (8.29) s(T2)−s(T1)= T2
T1
cp
T dT ≈cpln(T2/T1), (8.30) where the approximate expressions hold ifcpis a constant.Figure 8–6shows sketches ofh andsas a function ofTalong the linep=p1inFigure 8–4.
Along the linep=p2, there are two phase transitions, from S to L and from L to V, soμ has a discontinuity of slope at each transition, andhandshave jumps at each transition.
Along the linep=p3, there is only one phase transition, becausep3 >pcand there is no distinction between liquid and vapor above the critical pressure.
Next, we choose three constant temperaturesT1,T2, andT3, as indicated inFigure 8–7.
Along a line of constant T, μ is continuous and within a single phase, according to Eq. (8.7), it has a slope of v.Figure 8–8is a sketch ofμ versuspat T=T1. We observe the discontinuity of slope as the vapor-solid coexistence curve is crossed.
h
TSV T Δh
S
s
TSV T Δs
S
FIGURE 8–6 Sketches of the enthalpyhper mole and entropysper mole as a function ofTalong the linep=p1
inFigure 8–4. The full line corresponds to the stable solid and vapor phases. The stable phase is solid (S) forT≤TSV
and vapor (V) forT ≥ TSV. The dashed lines are extrapolations into unstable regions of superheated solid and supercooled vapor. The jumphin enthalpy is the latent heat of vaporization per mole and the jumpsis the entropy of vaporization per mole. These jumps are related byh=Tsso there is no jump inμ, consistent with Figure 8–5.
p
T Tt, pt
Tc, pc
S L
V
T1 T2 T3
FIGURE 8–7 Constant temperature pathsT1,T2, andT3on a phase diagram for the monocomponent system of Figure 8–1. The chemical potentialμis continuous along these paths, but its slope,v, changes asp crosses a coexistence curve.
à
pSV p V
S
FIGURE 8–8 Sketch of the chemical potentialμas a function ofpalong the lineT =T1inFigure 8–7. The full line corresponds to the stable solid and vapor phases. The dashed lines are extrapolations into unstable regions of superheated solid and supercooled vapor, intended to emphasize the discontinuity of slope of the full line at the phase transition. The stable phase is vapor (V) forp≤pSVand solid (S) forp≥pSV.
118 THERMAL PHYSICS
h Δh
pSV p V
S
s Δs
pSV p V
S
FIGURE 8–9 Sketches of the enthalpyhper mole and the entropysper mole as a function ofpalong the line T=T1inFigure 8–7. The full line corresponds to the stable solid and vapor phases. The stable phase is vapor (V) for p≤pSVand solid (S) forp≥pSV. The dashed lines are extrapolations into unstable regions of superheated solid and supercooled vapor. The jumphis the latent heat of vaporization and the jumpsis the entropy of vaporization.
These jumps are related byh=Tsso there is no jump inμ.
The behaviors of h and s versus p within a single phase can be ascertained from Eqs. (8.26) and (8.28) which along a line of constantTlead to
h(p2)−h(p1)= p2 p1
v(1−Tα)dp; (8.31)
s(p2)−s(p1)= p2
p1
−vαdp. (8.32)
For an ideal vapor,Tα=1 which gives
h(p2)−h(p1)=0; s(p2)−s(p1)= p2 p1
−v
Tdp= −Rln(p2/p1). (8.33) For the solid phase,Tα1, so for constantvwe have5
h(p2)−h(p1)≈v(p2−p1); s(p2)−s(p1)≈0. (8.34) Figure 8–9shows sketches ofhandsas functions ofpalong a lineT =T1inFigure 8–7.
Along the lineT=T2inFigure 8–7, there are two phase transitions, from V to L and from L to S, soμhas a discontinuity of slope at each transition andhandshave jumps at each transition. Along the lineT=T3, the liquid-vapor phase transition is absent becauseT3>
Tcand there is no distinction between liquid and vapor above the critical temperature.