H.2 Euler-Maclaur in Sum Formula
12.1 Reactions at Constant Volume or Pressure
Chemical reactions are typically carried out either at constant volume or at constant pressure. Those involving gases can usually be carried out easily at constant volume because the gases can be contained in a strong and nearly inert solid container. Then the workW=0, so the change in internal energy of the gases is
ifU=Q, constant volume, chemically closed, (12.3) where the heat Q is positive if added to the gases and negative if extracted from the gases. If the reaction vessel is thermally insulated, the reaction will result in a change of temperature that can be measured. For example, a bomb calorimeteris a rigid vessel with a known heat capacityCcalthat is large compared to the heat capacity of the gases undergoing reaction. Typically it is filled with oxygen at high pressure and some fuel that is burned to completion during the reaction. If the calorimeter is well insulated from its surroundings and its temperature changes byifT, thenQ= −CifT, whereCis the heat capacity of the calorimeter and the gases. To extent that the heat capacity of the gases can be neglected,CcalifTrepresents the energy that is converted from chemical bond energy as a result of the reaction.
Of great practical importance, however, are chemical reactions that are carried out such that the only work done is against a constant external pressurepext. In such reactions, there is a volume change ifV and there is no attempt to impose the constraint of constant volume, which might be very difficult if only condensed phases are involved. Moreover, the atmosphere might provide the constant external pressure in industrial reactions. The work done by the system will then beW=pextifVand from the first law we will have
ifU+pextifV=Q. (12.4)
If the pressurep= pextin the initial and final states of the system, we can introduce the enthalpyH =U+pV in which case Eq. (12.4) takes the form
ifH=Q, constant pressure, chemically closed, (12.5) whereQis the heat added to the reacting system. Thus the enthalpyHplays the same role at constant pressure as the internal energyUplays at constant volume. In general, one can regard the enthalpy to be a function of its natural variables, in which case
dH=TdS+Vdp+
i
μidNi. (12.6)
However, for practical purposes it is more convenient to use the temperature instead of the entropy, which results in
dH= ∂H
∂T
p,Ni
dT+ ∂H
∂p
T,Ni
dp+
i
H¯idNi, (12.7)
where the quantitiesH¯i are the partial molar enthalpies. We recognize (∂H/∂T)p,Ni = T(∂S/∂T)p,Ni = Cp as the heat capacity at constant pressure. Furthermore, regarding S to depend onT,p,Ni, we readily establish thatH¯i = μi−TS¯iand
∂H/∂p
T,Ni = V + T
∂S/∂p
T,Ni. A Maxwell relation based on the differential dG= −SdT+Vdp+
iμidNi
readily yields
∂S/∂p
T,Ni = −(∂V/∂T)p,Ni = −Vα, whereαis the coefficient of thermal expansion. Thus Eq. (12.7) can be written
dH=CpdT+V(1−αT)dp+
i
H¯idNi. (12.8)
For the chemically closed systems we are considering, Eq. (12.8) takes the form dH=CpdT+V(1−αT)dp+(
i
νiH¯i)dN˜. (12.9) Since T and p are intensive variables, the Euler equation for the enthalpy (see Eq. (5.101)) is just
H =
i
NiH¯i. (12.10)
We emphasize that Eq. (12.10) holds as a function ofp,T, andNi, provided that theH¯i are evaluated atp,T and the corresponding composition. At any stage of the reaction, Ni=Ni0+νiN˜, whereNi0is the initial value ofNi. The Euler equation (12.10) becomes
H(T,p,Ni)=
i
(Ni0+νiN˜)H¯i, (12.11) where it is understood that theH¯iare to be evaluated at the corresponding composition, temperature, and pressure.
Example Problem 12.1. For the chemical reaction given by Eq. (5.122), namely C+(1/2)O2→ CO, assume initially that the mole numbers areNC0 = 3,NO0
2 = 1, andNCO0 = 2. If conditions are such that the reaction goes to the right until one of the reactants is completely used, what is the value ofN˜finaland how many moles of each component will there be? Answer the same question under different conditions for which the reaction goes to the left until all of the CO is used.
Solution 12.1. The stoichiometric coefficients νi for C, O2, and CO are−1,−1/2, and 1, respectively. For either the forward or backward reaction we haveNC=3− ˜N,NO2=1−(1/2)N,˜ andNCO = 2+ ˜N. The reaction can go to the right untilN˜final = 2 = ˜Nmaxin which case NC = 1,NO2 = 0, andNCO = 3. The reaction can go to the left untilN˜final = −1 = ˜Nminin which caseNC=4,NO2=3/2, andNCO=0. The actual direction of the reaction and the extent of reaction will depend on the conditions under which the reaction is carried out, particularly the temperature. For conditions to be discussed below, the reaction may reach equilibrium at some valueN˜min≤ ˜Nfinal≤ ˜Nmax.
170 THERMAL PHYSICS
12.1.1 Heat of Reaction
According to Eq. (12.5), the heatQp = −Q liberated to the environment by the reacting system at constant pressurepis given by
−Qp=H(Tfinal,p,Ni0+νiN˜final)−H(Tinitial,p,Ni0). (12.12) ButQpis not a very useful way to characterize a reaction because it depends specifically on the initial conditions. A much more useful quantity is the derivative ofHwith respect to the progress variableN˜ at constant temperature and pressure, namely
H≡ −QN˜ :=
i
νiH¯i= ∂H
∂N˜
T,p
. (12.13)
This quantity is commonly called “the H of the reaction” but that is somewhat of a misnomer because it is a derivative. In particular,H should not be confused with−Qp
for a specific reaction, which is the difference in enthalpy between final and initial states given by Eq. (12.12). QN˜ = −H is the heatliberated by the reaction per unit change of the progress variable at constantpandT. Callen [2, p. 170] refers toHas theheat of reaction and suggests that it be evaluated near the equilibrium state; however, depending on conditions, a specific reaction might go to completion before the equilibrium state is reached. For QN˜ = −H > 0, the reaction is said to be exothermic whereas for QN˜ = −H < 0, the reaction is said to beendothermic.3 InSection 12.3we will relate Hto theGof the reaction.
For the special but often treated case for which the reactants and the products are not in solution, or if gaseous they form an ideal solution, one hasH¯i=Hi(p,T), whereHi(p,T) is the enthalpy per mole of the respective pure component. This follows for a solution of ideal gases because the chemical potentials
μi(T,p,Xi)=μi(T,p)+RTlnXi, (12.14) whereμi(T,p)corresponds to the pure component andXiis the mole fraction. Note that the total pressure ispand the partial pressure ispi=pXi. Thus,
H¯i=∂(μi(T,p,Xi)/T)
∂(1/T) =∂(μi(T,p)/T)
∂(1/T) =Hi(T,p), (12.15) so there is no heat of mixing for an ideal solution. Under these conditions, the initial and final states can be expressed in terms of heterogeneous components and Eq. (12.13) becomes simply
H=
i
νiHi(T,p), heterogeneous components, (12.16) for which there is extensive tabulation of data as discussed in the next section.
3Unfortunately, various authors use different terminology. Kondepudi and Prigogine [14, p. 53] associate the quantity
∂U/∂N˜
T,V =
iνiU¯iwith endothermic and exothermic reactions. Lupis [5, p. 10] and Kondepudi and Prigogine [14, p. 52] treatHfor the case in which the constituents are not in solution.