Article pubs.acs.org/jchemeduc
A Simple Method To Calculate the Temperature Dependence of the Gibbs Energy and Chemical Equilibrium Constants Francisco M. Vargas* Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas 77005, United States S Supporting Information *
ABSTRACT: The temperature dependence of the Gibbs energy and important quantities such as Henry’s law constants, activity coefficients, and chemical equilibrium constants is usually calculated by using the Gibbs−Helmholtz equation. Although, this is a well-known approach and traditionally covered as part of any physical chemistry course, the required mathematical treatment for solving the equation can be tedious and prone to mistakes especially when polynomial expressions for the heat capacities are used. In this article, an alternative approach is introduced in which a simple algebraic equation can provide quick and accurate results for the temperature dependence of the Gibbs energy and chemical equilibrium constants. The comparison against the conventional methods and values reported in the literature validate the effectiveness of the proposed equation. KEYWORDS: Second-Year Undergraduate, Upper-Division Undergraduate, Physical Chemistry, Problem Solving/Decision Making, Thermodynamics
T
G G1 − 0 =− T1 T0
he calculation of changes of the Gibbs energy (G) can give information on whether a process at fixed pressure and temperature is spontaneous (ΔG < 0), nonspontaneous (ΔG > 0), or in equilibrium (ΔG = 0). The mathematical definition of the Gibbs energy is based on the Legendre transform applied to the fundamental property relation of enthalpy:1
⎛ ∂H ⎞ ⎟ G = H − S⎜ ⎝ ∂S ⎠ P
T
Cp dT ) dT
0
(5)
(6)
⎡ ⎛1 ⎛ T ⎞ G T G1 1⎞ − 0 = −⎢ −H0⎜ − ⎟ + a⎜ln 1 + 0 − 1⎟ ⎢⎣ T1 T0 T0 ⎠ T1 ⎝ T1 ⎝ T0 ⎠
where H, S, and T are the enthalpy, entropy, and temperature, respectively. From eq 2, it can be deduced that G is associated with changes in pressure and temperature (measurable properties of a system). Eq 3 is the fundamental property relation for the Gibbs energy of a pure substance: (3)
For an isobaric process, the temperature dependence of Gibbs energy is usually calculated using the Gibbs−Helmholtz (GH) equation:
+
2 2 T2 ⎞ T3 ⎞ b⎛ c ⎛ T − 3T0 ⎜T1 − 2T0 + 0 ⎟ + ⎜ 1 + 0⎟ 2⎝ T1 ⎠ 3⎝ 2 T1 ⎠
+
3 3 T 4 ⎞⎤ d ⎛ T1 − 4T0 ⎜ + 0 ⎟⎥ 4⎝ 3 T1 ⎠⎥⎦
(7)
From this equation, ΔG can be calculated after some additional algebraic manipulation. However, this procedure is tedious and prone to calculation mistakes. The extension of the Gibbs−Helmholtz equation to calculate the temperature dependence of chemical equilibrium constants results in the well-known van’t Hoff equation (vHE):
(4)
By defining reference states, Go and To, the effect of temperature on the Gibbs energy can be calculated by a double integration, according to eq 5: © 2014 American Chemical Society and Division of Chemical Education, Inc.
∫T
where a, b, c, and d are constants defined for a given substance. The solution to the GH equation, eq 5, can be expressed as eq 7:
(2)
H ∂ ⎛⎜ G ⎞⎟ =− 2 ⎝ ⎠ ∂T T T
0
1 (H 0 + T2
Cp = a + bT + cT 2 + dT 3
This leads to the following well-known identity for the Gibbs energy:
d G = V dP − S d T
T1
Assuming that the isobaric heat capacity is a polynomial function of the temperature:
(1)
G = H − TS
∫T
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ΔHrxn ∂ ⎛ ΔGrxn ⎞ ⎜ ⎟=− ∂T ⎝ RT ⎠ RT 2
ΔGrxn,1 RT1
−
ΔGrxn,0
⎡K ⎤ ln⎢ 1 ⎥ = + ⎣ K0 ⎦
RT0
∫T
T1
and a finite change in this property can be expressed as (8)
=−
∫T
T1
ΔHrxn(T ) RT 2
0
ΔHrxn(T ) RT 2
0
dT
(9)
dT (10)
o ΔGrxn,0
−
RT0 Δd 3 T0 + 12R
ΔH =
ΔS =
T1
Cp dT = Cp̅ (T1 − T0) (constant P)
T1
Cp
0
T
∫T
dT = Cp̃ ln
T1 (constant P) T0
(18)
(19)
where C̅ p corresponds to the average isobaric heat capacity and C̃ p is a logarithmic average, defined according to eqs 20 and 21, respectively.
(11b)
T
Cp̅ =
∫T 1 Cp dT o
T1 − T0 b
(11c)
=
c
d
a(T1 − T0) + 2 (T12 − T02) + 3 (T13 − T03) + 4 (T14 − T04) T1 − T0 (20) T C
Cp̃ =
∫T 1 Tp dT o
ln
=
(12)
T1 T0
( ) + b(T − T ) + (T ln( )
a ln
Although eq 12 is simple to use, it might lead to a significant error in the calculation of the chemical equilibrium constant K1, if the temperature difference, T1 − T0, is large. The objective of this article is to present an alternative approach to calculate Gibbs energy and chemical equilibrium constants as a function of the temperature in a simple but also accurate form. Also, this work aims to enhance the learning experience of this topic, by including concepts related to state functions, reference conditions, Gibbs energy, and absolute entropy. This novel methodology can be introduced in a physical chemistry or thermodynamics course in conjunction with the traditional Gibbs−Helmholtz and van’t Hoff equations.
T1 T0
1
0
c 2
2 1
d
− T02) + 3 (T13 − T03)
T1 T0
(21)
Substitution of eqs 18 and 19 into eq 17 leads to eq 22, which allows the direct calculation of the change of the Gibbs energy as a function of the temperature. This equation is the foundation of the work presented in this article: ̃ ΔG = (Cp̅ − S0)(T1 − T0) − TC 1 p ln
T1 T0
(22)
where T0 and T1 are the initial and final temperatures, respectively, and S0 is the molar absolute entropy of the pure substance at temperature T0, and C̅ p and C̃ p are given by eqs 20 and 21, respectively. The values for C̅ p and C̃ p for selected species are available online as Supporting Information. The temperature dependence of these parameters is also presented in Figure 1. From Figure 1 it is clear that C̅ p is always greater than C̃ p and that the difference between these average values increases with increasing temperature and with the complexity and size of the molecule. For simple molecules, and for liquid and solids, C̅ p ≈ C̃ p and the values are less sensitive to changes in the temperature.
PROPOSED METHODS
Simple Method To Calculate the Temperature Dependence of the Gibbs Energy
Starting from eq 2, an infinitesimal change in the Gibbs energy can be defined as
dG = dH − d(TS)
∫T
0
However, because of the significant effort that is required to evaluate eqs 11a−c, it is a common practice to neglect the temperature dependence of the enthalpy of reaction to get an estimate of the equilibrium constant at temperature T1. If this is the case, Δa = Δb = Δc = Δd = 0 and the vHE can be approximated as
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(17)
Substituting the definitions for changes in enthalpy and entropy at constant pressure:
J Δa Δb Δc 2 ln T0 + T0 + T0 + 2R 6R RT0 R
⎡K ⎤ ΔHrxn ⎛ 1 1⎞ ln⎢ 1 ⎥ = − ⎜ − ⎟ R ⎝ T1 T0 ⎠ ⎣ K0 ⎦
(16)
= ΔH − S0ΔT − T1ΔS
where
I=
(15)
ΔG = ΔH − (S0(T1 − T0) + T1(S1 − S0))
(11a)
Δb 2 Δc 3 Δd 4 T0 − T0 − T0 2 3 4
ΔG = ΔH − (TS 1 1 − T0S0)
ΔG = ΔH − (TS 1 1 − T0S0 + TS 1 0 − TS 1 0)
J Δa Δb Δc 2 Δd 3 T1 − I T1 + T1 + + ln T1 + RT1 R 2R 6R 12R
o J = ΔHrxn,0 − ΔaT0 −
(14)
where subscripts 0 and 1 stand for the initial and final states, respectively. It is possible to manipulate the expression algebraically, by introducing a new term, T1S0, to get
where ΔGrxn and ΔHrxn are the Gibbs energy and enthalpy of reaction, respectively, and K0 and K1 are the chemical equilibrium constants at temperatures T0 and T1, respectively. For the most accurate calculations, the enthalpy of reaction should be considered a function of the temperature by using the polynomial expressions for the heat capacities of reactants and products as shown in eq 6. Elliot and Lira2 show that the solution for eq 10 can be expressed as ln K1 = −
ΔG = ΔH − Δ(TS)
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Writing similar equations for components B, C, and D and also using the fact that at equilibrium ΔG = 0: yP i 1
∑ υiRT1 ln
=
P0
i
⎡
∑ υi⎢−Gi(T0,P0) − (Cp̅ ,i − Si ,0) i
⎣
̃ (T1 − T0) + TC 1 p , i ln
T1 ⎤ ⎥ T0 ⎦
(27)
where yi is the mole fraction of species i, υi is the stoichiometric number and by convention, it is positive for products and negative for reactants. By algebraic manipulation, eq 27 is transformed into eqs 28 and 29a: ⎛ yP ⎞υi i 1 o ∑ RT1 ln⎜ ⎟ = −ΔGrxn,0 − ⎝ P0 ⎠ i
Figure 1. Comparison between the average heat capacities C̅ p (black line) and C̃ p (gray line) calculated using eqs 20 and 21, respectively, as functions of the final temperature, T1, for selected species. The polynomial expressions for Cp are taken from Elliot and Lira2 and T0 = 298 K.
1 o {−ΔGrxn,0 − RT1
(23)
∑ υi(ΔGi)P }
(28)
(29a)
i
T1 T0
(29b)
where (ΔGi)p is the change of Gibbs energy with temperature for component i at constant pressure, and K1 is the chemical equilibrium constant at temperature T1, which is defined as ⎛ yP ⎞νi i 1 K1 = ∏ ⎜ ⎟ P i ⎝ 0 ⎠
Simple Method To Calculate the Temperature Dependence of Chemical Equilibrium Constants
(30)
In this case, yC P1 γ
The proposed method to calculate the temperature dependence of the Gibbs energy can be readily extended to the analysis of a reactive system at equilibrium. Consider a hypothetical reaction in gaseous phase in which the behavior of the species can be assumed ideal:
K1 =
yD P1 δ
( )( ) ( )( ) P0
P0
yA P1
α
yB P1 β
P0
(31)
P0
because P0 = 1 bar,
(24)
In this case, for each substance, the Gibbs energy can be calculated at any pressure and temperature (T1, P1), starting from a reference state (T0, P0). For instance, the Gibbs energy for substance A in a mixture of ideal gases is given by
K1 =
(yC P1)γ (yD P1)δ (yA P1)α (yB P1)β
(32)
If eq 32 is used, it is important to keep in mind that the partial pressures must be expressed in the same units as P0. The value for ΔGorxn,0 is calculated from the Gibbs energies of formation for the different species, at the same reference conditions.
(25)
o ΔGrxn,0 =
where yA is the mole fraction of A in the gaseous mixture. Note that SA,0 is the absolute entropy of A at P0 and T0. In this case, the standard state for every component is taken as pure ideal gas at P0 = 1 bar and T0 = 298 K. For the reactive system at equilibrium (i.e., final state),
∑ υiGf ,i(T0 , P0) = ∑ υiGof ,i i
i
(33)
The procedure developed here, that is, eqs 29a,b, represents an alternative approach to the solution of existing vHE, eq 10.
■
APPLICATION EXAMPLES The following examples explain the procedure to calculate the temperature dependence of the Gibbs energy and the chemical equilibrium constants, using different methods.
ΔGrxn(T , P) = γGC(T , P) + δG D(T , P) − αGA (T , P) − β G B (T , P )
T1 ⎤ ⎥ T0 ⎦
̃ (ΔGi)P = (Cp̅ , i − Si ,0)(T1 − T0) − TC 1 p , i ln
where Cp can be calculated as an average value within the temperature interval (e.g., Cp = C̅ p), or for small changes in temperature, it can be taken as the heat capacity at the reference temperature, Cp,0, which is usually readily available in the literature.
GA (T1 , P1) = GA (T0 , P0) + (CP̅ ,A − SA,0)(T1 − T0) y P1 T1 ̃ − TC + RT1 ln A 1 P ,A ln T0 P0
⎣
with
Consequently, for simple molecules in the gaseous state and for liquids and solids:
α A (g) + β B(g) ⇄ γ C(g) + δ D(g)
i
̃ (T1 − T0) − TC 1 p , i ln
ln K1 =
T ΔG = (Cp − S0)(T1 − T0) − CpT1 ln 1 T0
⎡
∑ υi⎢(Cp̅ ,i − Si ,0)
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Table 1. Thermodynamic Data for Species Involved in Ammonia Synthesis Reactiona Cp = a + bT + cT2 + dT3, J mol−1 K−1
a
ΔHof / −1
ΔGof / −1
So/ J mol−1 K−1
a
b
c
d
C̅ p at 400 K/ J mol−1 K−1
C̃ p at 400 K/ J mol−1 K−1
191.61 130.68 192.45
31.15 27.14 27.31
−1.357 × 10−2 9.274 × 10−3 2.383 × 10−2
2.680 × 10−5 −1.381 × 10−5 1.707 × 10−5
−1.168 × 10−8 7.645 × 10−9 −1.185 × 10−8
29.19 29.01 37.21
29.19 29.01 37.13
Species
kJ mol
kJ mol
N2 (g) H2 (g) NH3 (g)
0 0 − 46.11
0 0 − 16.45
ΔHof , ΔGof and So are taken from ref 3. The coefficients for Cp are taken from ref 2.
Example 1: Change in Gibbs Energy with Temperature
ΔG = G1 − G0 = (35.06 − 192.45)(400 − 298) 400 − (35.06)(400)ln = −20.18 kJ mol−1 298
Calculate the change in Gibbs energy when 1 mol of gaseous ammonia is heated at constant pressure P = 1 bar from T0 = 298 K to T1 = 400 K. Assume that gaseous ammonia at these conditions behaves as an ideal gas. The expression for the heat capacity of ammonia is reported in literature2 as
In this case, the error produced by the approximation C̅ p = C̃ p = Cp,0 is less than 0.1%. Also, eq 35 gives the same result (i.e., ΔG = −20.18 kJ mol−1) when Cp,0 is used.
Cp = 27.31 + 2.383 × 10−2T + 1.707 × 10−5T 2 − 1.185 × 10−8T 3
Example 2: Chemical Equilibrium Constant at a Given Temperature
(34) −1
Calculate the chemical equilibrium constant for the formation of ammonia from its elements, in a gaseous phase reaction at T = 400 K, assuming that all the species behave as ideal gases. The chemical reaction that takes place in this case is
−1
where Cp is given in units of J mol K and T in K. Solution via the Gibbs−Helmholtz Equation. The Gibbs energy of formation of gaseous ammonia at 298 K is −16.45 kJ mol−1 and the absolute entropy, S0, at the same temperature is 192.45 J mol−1 K−1.3 Thus, H = G + TS = −16,450 + (298) (192.45) = 40,900 J mol−1. This method makes use of the integrated form of the Gibbs−Helmholtz equation, eq 7,
N2(g) + 3H 2(g) ⇄ 2NH3(g)
and relevant thermodynamic data for this reaction is given in Table 1. The values of C̅ p and C̃ p are taken from the Supporting Information, and are calculated using eqs 20 and 21 with the coefficients for Cp from ref 2. More information on thermophysical properties can be found in ref 4. Solution via Method 1. Full Solution of van’t Hoff Equation. Applying the solution to the vHE proposed by Elliot and Lira,2 with eqs 11a−c): Δa = −57.95, Δb = 3.341 ×10−2, Δc = 4.877 × 10−5, and Δd = −3.496 × 10−8; ΔHorxn,0 = −92,220 J, ΔGorxn,0 = −32,900 J, J = −76,795.7 J, I = −21.32,
ΔG = G1 − G0 ⎡ ⎛T ⎞ ⎛1 1⎞ = G0⎜ 1 − 1⎟ − T1⎢ −H0⎜ − ⎟ ⎢⎣ T0 ⎠ ⎝ T0 ⎠ ⎝ T1 ⎛ T ⎞ T T2 ⎞ b⎛ + a⎜ln 1 + 0 − 1⎟ + ⎜T1 − 2T0 + 0 ⎟ T1 T1 ⎠ 2⎝ ⎝ T0 ⎠ +
2 2 T 3 ⎞ d ⎛ T 3 − 4T03 T 4 ⎞⎤ c ⎛ T1 − 3T0 ⎜ + 0⎟+ ⎜ 1 + 0 ⎟⎥ 3⎝ 2 T1 ⎠ 4⎝ 3 T1 ⎠⎥⎦
−76,795.7 −57.95 + ln(400) (8.314)(400) 8.314 3.341 × 10−2 4.877 × 10−5 + (400) + 2(8.314) 6(8.314) −3.496 × 10−8 2 (400) + (400)3 + 21.32 12(8.314)
ln K1 = −
(35)
Substituting the known values, G0 = −16,450 J mol−1, T0 = 298 K, T1 = 400 K, a = 27.31, b = 2.383 × 10−2, c = 1.707 × 10−5, and d = −1.185 × 10−8 into the eq 35, ΔG = G1 − G0 = −20.20 kJ mol−1. Solution via the New Equation. Using the new approach proposed in this work, the Gibbs energy is directly calculated by applying eq 22. T1 ̃ ΔG = (Cp̅ − S0)(T1 − T0) − TC 1 p ln T0
(36)
ln K1 = 3.58 ⇒ K1 = 36.04
Solution via Method 2. Approximated Solution of van’t Hoff Equation. In this method, the effect of temperature on the enthalpy of reaction is neglected, and the chemical equilibrium constant is obtained by solving eq 12:
(22)
The values of C̅ p = 37.2 J mol−1 K−1 and C̃ p = 37.1 J mol−1 K−1 are taken from Table A.1 (in the Supporting Information), and S0 = 192.45 J mol−1 K−1 as in the previous method. With this information, the change of the Gibbs energy can be readily obtained: ΔG = G1 − G0 = −20.20 kJ mol−1. Note that the values obtained by the two methods are identical but the second method offers a simpler procedure. When the temperature dependence of the heat capacities is unknown or the interest is on a rough estimate of the changes of Gibbs energy, it can be assumed that C̅ p = C̃ p = Cp,0, where Cp,0 is the heat capacity at 298 K. For ammonia, Cp,0 = 35.06 J mol−1 K−1.3
⎡K ⎤ ΔHrxn ⎛ 1 1⎞ ln⎢ 1 ⎥ = − ⎜ − ⎟ R ⎝ T1 T0 ⎠ ⎣ K0 ⎦ o o ⎛ −ΔGrxn,0 ΔHrxn 1 1⎞ ⇒ ln K1 = − ⎜ − ⎟ RT0 R ⎝ T1 T0 ⎠ o ΔHrxn,0 = −92,220 J
ln K1 =
o ΔGrxn,0 = − 32,900 J
−32,900 −92,220 ⎛ 1 1 ⎞ ⎜ ⎟ − − (8.314)(298) 8.314 ⎝ 400 298 ⎠
ln K1 = 3.78 ⇒ K1 = 44.14 399
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Figure 2. Comparison of different methods to calculate the temperature dependence of equilibrium constant K1.
Solution via Method 3. New Equation (C̅ p ≠ C̃p and Both Are Functions of T). Applying eq 29b to the three species, with the information provided in Table 1: (ΔGN2) = −20,003.9 J mol−1, (ΔGH2) = −13,786.2 J mol−1, (ΔGNH3) = −20,206.5 J mol−1. And substituting the values in eq 29a with ΔGrxn,0 = −32,900 J, one can readily get ln K1 =
ln K1 =
1 {+32,900 − [(2)( −20,215.9) (8.314)(400) + 20,003.9 + (3)(13,786.2)]}
ln K1 = 3.60 ⇒ K1 = 36.56
Solution via Method 3b. New Equation (C̅ p = C̃p= Cp,0 = Constant). In this case, the heat capacities are assumed to be independent of the temperature. Thus, C̅ p = C̃ p = Cp,0, where Cp,0 is obtained from evaluation of the heat capacity polynomials at T = 298 K. These values are 29.2, 28.9, and 35.6 J mol−1 K−1 for N2, H2, and NH3, respectively. Substituting these numbers in eq 29b: (ΔGN2) = −20,004.1 J mol−1, (ΔGH2) = −13,784.5 J mol−1, (ΔGNH3) = −20,190.5 J mol−1. And substituting the values in eq 29a, with ΔGorxn,0 = −32,900 J, one can readily get
1 {+32,900 − [(2)( −20,206.5) (8.314)(400) + 20,003.9 + (3)(13,786.2)]}
ln K1 = 3.59 ⇒ K1 = 36.36
Solution via Method 3a. New Equation (C̅ p = C̃ p and C̅p Is a Function of T). In this case, only one average heat capacity is required. C̅ p = C̃ p = ∫ T1 T0 (Cp dT)/(T1 − T0), (ΔGN2) = −20,003.9 J mol−1, (ΔGH2) = −13,786.2 J mol−1, (ΔGNH3) = −20,215.9 J mol−1. And substituting the values in eq 29a with ΔGorxn,0 = −32,900 J, one can readily get
ln K1 =
1 {+32,900 − [(2)( −20,190.5) (8.314)(400) + 20,004.1 + (3)(13,784.5)]}
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ln K1 = 3.58 ⇒ K1 = 36.07
Article
CONCLUSIONS AND PEDAGOGICAL IMPLICATIONS A new method to calculate the temperature dependence of the Gibbs energy has been developed. The proposed model enables an accurate and simple method for determining the changes in Gibbs energy as a function of the temperature. Furthermore, the procedure can be readily extended to the calculation of chemical equilibrium constants at any temperature. In the conventional van’t Hoff method, it is usually assumed that the enthalpy of reaction is independent of the temperature to facilitate the calculations, which can induce a significant error in the case of reactions occurring at high temperature and/or when the difference ΔCp is large. In the new model, the assumption of constant enthalpy of reaction is not required, and the simplicity of the equations is retained. Furthermore, as it was also proven in this article, the new equation gives the same level of accuracy as the full van Hoff equation, but because the calculation procedure is greatly simplified, it is less prone to calculation mistakes.
From all the five methods tested, the approximated solution of vHE (method 2) shows the greatest deviation from the value obtained using the full solution of vHE (method 1). The equation proposed in this work and its variants (methods 3, 3a, and 3b) give very similar results to the full solution of vHE (method 1). In fact, the small difference between methods 1 and 3 is due to round-off errors. These two examples illustrate the application of the different methods to calculate the temperature dependence of the chemical equilibrium constants, and it is clear that although the solution of the full vHE offers an accurate result, the effort to solve the necessary equations is also significant. The model that is proposed in this work offers the same level of accuracy, but with a simpler procedure.
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■
VALIDATION OF THE PROPOSED EQUATION To provide a full validation of the proposed model and a more detailed analysis of the limitation of the other methods, four different reactions in a wide range of temperatures (273−1500 K) have been studied. The full solution of the vHE (method 1) for the different cases is used as a reference (i.e., exact solution) and the simplified solution of vHE (method 2), the new equation (method 3) as well as two of its variants (methods 3a and 3b) are also tested in this study. The results are presented in Figure 2. In all the cases, the new equation (method 3, represented by a continuous black line in Figure 2), perfectly matches with the results obtained from the full vHE (method 1, black circles) in the entire temperature range, which is expected, because the two methods are mathematically equivalent. The simplified solution of vHE (method 2, gray line) works well for the ethylene hydration reaction, but increasingly underperforms as the temperature increases for the other three reactions. Method 2 works particularly well when the ΔCp is small, which is the case for the ethylene hydration reaction. As expected, the error increases with increasing temperature and for larger magnitudes of ΔCp. The sign of ΔCp defines the type of deviation in the calculation. Positive values of ΔCp cause underprediction of the value of lnK1, whereas negative values of ΔCp cause overprediction. In method 3a, it is assumed that the two average heat capacities are equal and that C̅ p is a function of the temperature. The results shown in Figure 2 as a dashed line validate that this method gives a good approximation, while it reduces the effort in calculating the average heat capacity values. In method 3b, it is assumed that the heat capacities are constants, independent of the temperature (C̅ p = C̃ p = Cp,0) where Cp,0 is the heat capacity at 298 K. Method 3b, represented in Figure 2 by a dotted line, gives a better prediction than the simplified solution of vHE (method 2) for cases where ΔCp is large, and the values of the heat capacities do not change significantly with the temperature. As presented in Figure 1, this is especially the case for simple molecules, such as H2, N2, O2, NO, CO. In fact, a combination of methods 3a and 3b is feasible, in which C̅ p = C̃ p = Cp,0 can be used for simple molecules involved in the reaction, whereas a full calculation is done to determine C̅ p and C̃ p, from eqs 20 and 21, for more complex molecules. The worst result obtained by method 3b is for the ethylene hydration reaction because both ethylene and ethanol double the values of their heat capacities from 273 to 1500 K.
ASSOCIATED CONTENT
* Supporting Information S
Heat capacity table. This material is available via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The author gratefully acknowledges support from Tecnológico de Monterrey, through Cátedra de Energiá Solar y Termociencias (Grant CAT-125). The support of Shyam “Benny” Kadali and Sameer Punnapala in early versions of this manuscript is also greatly appreciated. The author also thanks the anonymous reviewers for their careful assessment, which helped to improve this article. In particular, section 4 is largely the result of addressing the constructive feedback of one of the referees.
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REFERENCES
(1) Beegle, B. L.; Modell, M.; Reid, R. C. AIChE J. 1974, 20, 1194− 1200. (2) Elliot, J. R.; Lira, C. T. Introductory Chemical Engineering Thermodynamics, 2nd ed.; Prentice Hall: New York, 2009; pp 652−657. (3) Atkins, P.; de Paula, J. Physical Chemistry, 9th ed.; W.H. Freeman: and Co.: New York, 2009; pp 918−923. (4) NIST Chemistry WebBook. http://webbook.nist.gov/chemistry/ (accessed Feb 2014).
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