A mathematical approach to chemical equilibrium theory for gaseous systems IV: a mathematical clarification of Le Chatelier’s principle

Abstract

Chemical equilibrium is usually discussed via a thermodynamic treatment but this does not automatically provide enough mathematical tools to be successful. A complementary mathematical approach was developed in a series of previous papers to reveal the inner logic in equilibrium shift for gaseous systems. Le Chatelier’s principle is reconsidered with this approach and a system has been developed in order to fully address the application of this principle and the dangers of using it without due consideration. In this study it is demonstrated that, more often than not, real conceptual understanding can only be achieved through mathematical derivations which help to build a more rigorous and abstract understanding. The image of chemistry can be improved by introducing more mathematics into elementary chemistry. Real chemical education delves into curricular contents and offers deep insight into chemistry which provides crucial assistance to chemical teachers and benefits students.

Appendices

Appendices

1.1 Appendix 1: Another proof for Theorem 1 with gaseous species on both sides of the chemical equation

It has been pointed out previously [15] that there is a mistake in the Proof of Theorem 1 by Katz [10]. The proof below corrects that mistake. Since

$$\begin{aligned} n_1^2 v_2^2 +n_2^2 v_1^2 \ge 2v_1 v_2 n_1 n_2 \end{aligned}$$

(57)

We obtain

$$\begin{aligned} \frac{v_1^2 }{n_1 }+\frac{v_2^2 }{n_2 }\ge \frac{(v_1 +v_2 )^{2}}{n_1 +n_2} \end{aligned}$$

(58)

Equation 58 can also be obtained when Eq. 59 is valid.

$$\begin{aligned} \frac{n_2 }{n_1 }= & {} \frac{v_2 }{v_1 }>0 \end{aligned}$$

(59)

$$\begin{aligned} \frac{v_1^2}{n_1}+\frac{v_2^2 }{n_2 }= & {} \frac{(v_1 +v_2 )^{2}}{n_1 +n_2} \end{aligned}$$

(60)

$v_1$ and $v_2$ in Eq. 59 are either both positive or negative since $\hbox {n}_{1}$ and $\hbox {n}_{2}$ are both positive. If $v_1$ and $v_2$ have opposite signs, the equal sign should be excluded in Eq. 58. Equation 60 can also be derived from Schwarz inequality 61.

$$\begin{aligned} \sum _{i=1}^N {a_i^2 } \sum _{i=1}^N {b_i^2 } \ge \left( \sum _{i=1}^N {a_i b_i}\right) ^{2} \end{aligned}$$

(61)

Inequality 61 is equivalent to Eq. 62 [15].

$$\begin{aligned} \sum _{i=1}^N {\sum _{j>i}^N {(a_i b_j } } -a_j b_i )^{2}\ge 0 \end{aligned}$$

(62)

From Eq. 62 it is clear that the equals sign in Eq. 61 is valid when Eq. 63 is satisfied for all i and j. From Eq. 63 we know that the equals sign can be included in the Schwarz inequality 61 when the signs for $\hbox {a}_{\mathrm{i}}$ and $\hbox {b}_{\mathrm{i}}$ are the same, but not otherwise.

$$\begin{aligned} \frac{a_i }{b_i }=\frac{a_j }{b_j },\quad i,j=1,2,3,\ldots ,N \end{aligned}$$

(63)

For example Eq. 61 is valid together with Eq. 64, i.e. including the equal sign.

$$\begin{aligned} a_i >0,b_i >0,\quad i=1,2,3,\ldots ,N \end{aligned}$$

(64)

When the signs for $\hbox {a}_{\mathrm{i}}$ and $\hbox {b}_{\mathrm{i}}$ are mixed, i.e. if some $\hbox {a}_{\mathrm{i}}$ has the same sign as its counterpart $\hbox {b}_{\mathrm{i}}$ while another $\hbox {a}_{\mathrm{j}}$ has the opposite sign to its counterpart $\hbox {b}_{\mathrm{j}}$, Eq. 63 cannot be satisfied for all i and j, thus the equals sign in Eq. 61 should be excluded.

It can be demonstrated that Eq. 58 is one form of the Schwarz inequality shown by Eq. 61. If we define

$$\begin{aligned} a_i =\frac{\left| {v_i } \right| }{\sqrt{n_i }},\quad b_i =\sqrt{n_i} \end{aligned}$$

(65)

note $\hbox {n}_{\mathrm{i}}>0$. Equation 61 becomes

$$\begin{aligned} \sum _{i=1}^N {\left| {\frac{v_i }{\sqrt{n_i }}} \right| } ^{2}\sum _{i=1}^N {n_i } \ge \left( \sum _{i=1}^N {\left| {v_i } \right| } \right) ^{2} \end{aligned}$$

(66)

Rearranging Eq. 66 we obtain Eq. 67 which is in a similar form to Eq. 58.

$$\begin{aligned} \sum \limits _{i=1}^N {\frac{v_i^2 }{n_i }} \ge \frac{\left( \sum \nolimits _{i=1}^N \left| {v_i } \right| \right) ^{2}}{\sum \nolimits _{i=1}^N {n_i}} \end{aligned}$$

(67)

Equation 67 is obtained by Katz [10] using mathematical induction. Both Eqs. 68 and 69 conform to the result implied by Eq. 67.

$$\begin{aligned} \sum \limits _{i=1}^{N_p } {\frac{v_{p_i }^2 }{n_{p_i } }}\ge & {} \frac{\left( \sum \nolimits _{i=1}^{N_p } v_{p_i } \right) ^{2}}{\sum \nolimits _{i=1}^{N_p } {n_{p_i}}} \end{aligned}$$

(68)

$$\begin{aligned} \sum \limits _{i=1}^{N=N_r } {\frac{v_{r_i }^2 }{n_{r_i } }}\ge & {} \frac{\left( \sum \nolimits _{i=1}^{N=N_r } v_{r_i } \right) ^{2}}{\sum \nolimits _{i=1}^{N=N_r } {n_{r_i } } } \end{aligned}$$

(69)

If there are gaseous species on both sides of the chemical equation, we obtain Eq. 70 since $v_p$ and $v_r$ have opposite signs.

$$\begin{aligned} \sum \limits _{i=1}^{N=N_p +N_r } {\frac{v_i^2 }{n_i }} >\frac{\left( \sum \nolimits _{i=1}^{N=N_p +N_r } v_i\right) ^{2}}{\sum \nolimits _{i=1}^{N=N_p +N_r } {n_i } }=\frac{\Delta v^{2}}{n_T} \end{aligned}$$

(70)

The validity of Eq. 70 can best be demonstrated by comparing it with Eq. 67 which is consistent with Eqs. 68 and 69 and was proposed by Katz [10], by using Eq. 71, i. e. because of Eq. 71, the equal sign in Eq. 70 has been excluded when there are gaseous species on both side of chemical equation.

$$\begin{aligned} \sum _{i=1}^N {\left| {v_i } \right| } >\sum _{i=1}^N {v_i} \end{aligned}$$

(71)

Equation 70 conforms to Theorem 1 for the case when there are gaseous species on both sides of the chemical equation. It should be noted that Eq. 70 is correct in chemistry but not in mathematics since in the latter $\hbox {n}_{\mathrm{i}}$ can be negative.

1.2 Appendix 2: Relevant mathematics for some aspects of Theorem 2

From Eq. 3 we obtain Eq. 72 for adding species j into a chemical system.

$$\begin{aligned} Q_x (n_j +dn_j )=\frac{\left( 1+\frac{v_j dn_j }{n_j}\right) \prod \nolimits _{i=1}^N {n_i^{\nu _i } } }{\left( 1+\frac{\Delta vdn_j }{n_T }\right) n_T^{\Delta \nu } } \end{aligned}$$

(72)

If we ignore the change in denominator $\hbox {D}_{\mathrm{x}}$ in Eq. 72, we obtain Eq. 74.

$$\begin{aligned} \left. {Q_x (n_j +dn_j )} \right| _{D_x } =\left( 1+\frac{v_j dn_j }{n_j }\right) Q_x (n_j ) \end{aligned}$$

(73)

or

$$\begin{aligned} \left. {\Delta Q_x } \right| _{D_x } =\left. {\left[ Q_x (n_j +dn_j )-Q_x (n_j)\right] } \right| _{D_x } =\frac{v_j dn_j }{n_j }Q_x (n_j ) \end{aligned}$$

(74)

If we ignore the change in numerator $\hbox {N}_{\mathrm{x}}$ in Eq. 72, we obtain Eq. 76.

$$\begin{aligned} \left. {Q_x (n_j +dn_j )} \right| _{N_x }= & {} \frac{1}{\left( 1+\frac{\Delta vdn_j }{n_T }\right) }Q_x (n_j ) \nonumber \\= & {} \left( 1+\frac{\Delta vdn_j }{n_T}\right) ^{-1}Q_x (n_j )=\left( 1-\frac{\Delta vdn_j}{n_T}\right) Q_x (n_j )\qquad \end{aligned}$$

(75)

or

$$\begin{aligned} \Delta \left. {Q_x } \right| _{N_x } =\left. {\left[ Q_x (n_j +dn_j )-Q_x (n_j)\right] } \right| _{N_x } =-\frac{\Delta vdn_j }{n_T }Q_x (n_j ) \end{aligned}$$

(76)

Parts (a) and (b) of Theorem 2 can be discussed along with Eq. 72. The discussion below is relevant to corollary ii of Theorem 2. From Eqs. 74 and 76, $\left. {\Delta Q_x } \right| _{D_x }$ and $\Delta \left. {Q_x}\right| _{N_x}$ have the same sign when $v_j$ has an opposite sign to $\Delta v$, i.e. for the added species j on the side of chemical reaction with the smaller sum of coefficients. Thus the effects described in parts (a) and (b) shift the equilibrium in the same direction when $\frac{v_j }{\Delta v}<0$. But if $v_j$ and $\Delta v$ have the same sign, $\left. {\Delta Q_x } \right| _{D_x}$ and $\Delta \left. {Q_x } \right| _{N_x}$ have opposite signs and the effects described in parts (a) and (b) shift the equilibrium in opposite directions. $\left. {\Delta Q_x } \right| _{D_x}$ overrides $\Delta \left. {Q_x } \right| _{N_x }$ when Eq. 77 is satisfied, i.e. the effect specified in Theorem 2b is dominant.

$$\begin{aligned} \left| {\frac{v_j dn_j }{n_j }Q_x (n_j )} \right| >\left| {\frac{\Delta vdn_j }{n_T }Q_x (n_j )} \right| \end{aligned}$$

(77)

When both $v_j$ and $\Delta v$ are negative, we obtain Eqs. 78 and 79 from 77.

$$\begin{aligned} \frac{v_j }{n_j }<\frac{\Delta v}{n_T} \end{aligned}$$

(78)

or

$$\begin{aligned} x_j =\frac{n_j }{n_T }<\frac{v_j }{\Delta v}>0 \end{aligned}$$

(79)

Thus, as specified in Theorem 2ii, the effect described by Theorem 2b is dominant when $x_j <\frac{v_j }{\Delta v}>0$. Similarly, the effect described by Theorem 2a is dominant when $x_j >\frac{v_j }{\Delta v}>0$. When both $v_j$ and $\Delta v$ are positive, a similar discussion can be invoked and the result is specified by Theorem 2ii.

1.3 Appendix 3: A detailed mathematical account of conjugated variables

The common feature of conjugated variable pairs for $\hbox {x}_{\mathrm{j}}$ and $\hbox {n}_{\mathrm{j}},\hbox { C}_{\mathrm{j}}$ and $\hbox {n}_{\mathrm{j}}$, or P and V in Eqs. 80–82 [17] is that they contain a square term in

$$\begin{aligned} \left( {\frac{\partial x_j }{\partial \zeta }} \right) _{n_i } \left( {\frac{\partial Q_x }{\partial n_j }} \right) _{\zeta ,n_i } , \,\left( {\frac{\partial C_j }{\partial \zeta }} \right) _{V,n_i } \left( {\frac{\partial Q_C }{\partial n_j }} \right) _{V,\zeta }, \,\hbox {or}\, \left( {\frac{\partial P}{\partial \zeta }} \right) _{T,V} \left( {\frac{\partial Q_P }{\partial V}} \right) _{T,\zeta } , \end{aligned}$$

such as $(\nu _j -\Delta \nu x_j )^{2}$ in Eq. 80, $(\nu _j )^{2}$ in Eq. 81, and $(\Delta \nu )^{2}$ in Eq. 82.

$$\begin{aligned} \left( {\frac{\partial x_j }{\partial n_j }} \right) _{T,P,Q_x =K_x }= & {} \left( {\frac{\partial x_j }{\partial n_j }} \right) _\zeta -\left( {\frac{\partial x_j }{\partial \zeta }} \right) _{n_i } \frac{\left( {\frac{\partial Q_x }{\partial n_j }} \right) _{\zeta ,n_i } }{\left( {\frac{\partial Q_x }{\partial \zeta }} \right) _{n_i } } \nonumber \\= & {} \frac{1-x_j }{n_T }-\frac{\nu _j -\Delta \nu x_j }{n_T }\frac{Q_x \left( {\frac{\nu _j -\Delta \nu x_j }{n_i }} \right) }{\left( {\frac{\partial Q_x }{\partial \zeta }} \right) _{n_i^0 } }\ge 0 \end{aligned}$$

(80)

$$\begin{aligned} \left( {\frac{\partial C_j }{\partial n_j }} \right) _{T,V,Q_C =K_C }= & {} \left( {\frac{\partial C_j }{\partial n_j }} \right) _\zeta -\left( {\frac{\partial C_j }{\partial \zeta }} \right) _{V,n_i } \frac{\left( {\frac{\partial Q_C }{\partial n_j }} \right) _{V,\zeta } }{\left( {\frac{\partial Q_C }{\partial \zeta }} \right) _{V,n_i } } \nonumber \\= & {} \frac{1}{V}-\frac{\nu _j }{V}\frac{Q_C \frac{\nu _j }{n_j }}{Q_C \sum \limits _{i=1}^N {\frac{\nu _i^2 }{n_i }} }\ge 0 \end{aligned}$$

(81)

$$\begin{aligned} \left( {\frac{\partial P}{\partial V}} \right) _{T,Q_p =K_P }= & {} \left( {\frac{\partial P}{\partial V}} \right) _{T,\zeta } +\left( {\frac{\partial P}{\partial \zeta }} \right) _{T,V} \frac{\left( {\frac{\partial Q_P }{\partial V}} \right) _{T,\zeta } }{\left( {\frac{\partial Q_P }{\partial \zeta }} \right) _{T,V} } \nonumber \\= & {} -\frac{P}{V}-\frac{\Delta \nu P}{n_T }\frac{-\frac{\Delta \nu Q_P }{V}}{Q_P \sum \limits _{j=1}^N {\frac{\nu _j^2 }{n_j }} }\le 0 \end{aligned}$$

(82)

In Fig. 3, the shift from (1) to ($2'$) represents $\left( {\frac{\partial P}{\partial V}} \right) _{T,\zeta }$ in Eq. 82, while the shift from ($2'$) To ($3'$) is represented by the term $\left( {\frac{\partial P}{\partial \zeta }} \right) _{T,V} \frac{\left( {\frac{\partial Q_P }{\partial V}} \right) _{T,\zeta } }{\left( {\frac{\partial Q_P }{\partial \zeta }} \right) _{T,V} }$, and from (1) to ($3'$) is the resultant derivative $\left( {\frac{\partial P}{\partial V}} \right) _{T,Q_p =K_P}$. ($4'$) is a state inaccessible to the system which is indicated by $\le 0$ in Eq. 82.

Fig. 3

A pictorial representation of the derivatives in Eq. 82

On the other hand, $\hbox {x}_{\mathrm{k}}$ and $\hbox {n}_{\mathrm{j}}$ are not a pair of conjugated variables because there is not a square term in $\left( {\frac{\partial x_k }{\partial \zeta }} \right) _{n_j }\left( {\frac{\partial Q_x }{\partial n_j }} \right) _{\zeta ,n_i}$ as shown in Eq. 83.

$$\begin{aligned} \left( {\frac{\partial x_k }{\partial n_j }} \right) _{T,P,Q_x =K_x }= & {} \left( {\frac{\partial x_k }{\partial n_j }} \right) _\zeta -\left( {\frac{\partial x_k }{\partial \zeta }} \right) _{n_j } \frac{\left( {\frac{\partial Q_x }{\partial n_j }} \right) _{\zeta ,n_i } }{\left( {\frac{\partial Q_x }{\partial \zeta }} \right) _{n_i } } \nonumber \\= & {} \frac{-x_k }{n_T }-\frac{\nu _k -\Delta \nu x_k }{n_T }\frac{Q_x \left( {\frac{\nu _j -\Delta \nu x_j }{n_j }} \right) }{\left( {\frac{\partial Q_x }{\partial \zeta }} \right) _{n_i^0 } } \end{aligned}$$

(83)

A result similar to Eq. 82 can also be derived from thermodynamics [8]. For example, function A is defined in Eq. 84 for a system with entropy S.

$$\begin{aligned} A(S,V,\zeta )=\sum _i {\nu _i } \mu _i \end{aligned}$$

(84)

At equilibrium, we have

$$\begin{aligned} dA=\left( \frac{\partial A}{\partial S}\right) _{\zeta ,V} dS+\left( \frac{\partial A}{\partial V}\right) _{S,\zeta } dV+\left( \frac{\partial A}{\partial \zeta }\right) _{S,V} d\zeta =0 \end{aligned}$$

(85)

From Eq. 85 we obtain Eq. 86.

$$\begin{aligned} \left( {\frac{\partial \zeta }{\partial V}} \right) _{S,A} =-\frac{\left( {\frac{\partial A}{\partial V}} \right) _{S,\zeta } }{\left( {\frac{\partial A}{\partial \zeta }} \right) _{S,V}} \end{aligned}$$

(86)

Equation 88 is obtained from Eq. 87.

$$\begin{aligned} dU= & {} TdS-PdV+Ad\xi \end{aligned}$$

(87)

$$\begin{aligned} \left( {\frac{\partial P}{\partial \zeta }} \right) _{S,V}= & {} -\left( {\frac{\partial A}{\partial V}} \right) _{S,\zeta } \end{aligned}$$

(88)

From Eq. 86 and 88 we obtain Eq. 89 from $P=P[S,V,\zeta (S,V,A)]$.

$$\begin{aligned} \left( {\frac{\partial P}{\partial V}} \right) _{S,A}= & {} \left( {\frac{\partial P}{\partial V}} \right) _{S,\zeta } +\left( {\frac{\partial P}{\partial \zeta }} \right) _{S,V} \cdot \left( {\frac{\partial \zeta }{\partial V}} \right) _{S,A} \nonumber \\= & {} \left( {\frac{\partial P}{\partial V}} \right) _{S,\zeta } +\left( {\frac{\partial A}{\partial V}} \right) _{S,\zeta } \frac{\left( {\frac{\partial A}{\partial V}} \right) _{S,\zeta } }{\left( {\frac{\partial A}{\partial \zeta }} \right) _{S,V} } \end{aligned}$$

(89)

There is a square term $\left[ \left( {\frac{\partial A}{\partial V}} \right) _{S,\zeta }\right] ^{2}$ in Eq. 89. From Eq. 90 [17] it can be seen that the term $\left( {\frac{\partial P}{\partial \zeta }} \right) _{S,V} \cdot \left( {\frac{\partial \zeta }{\partial V}} \right) _{S,A}$ is positive in Eq. 89. It should be noticed that $\left( {\frac{\partial P}{\partial V}} \right) _{S,\zeta }$ is negative. Although it can be proved from thermodynamics that the signs for $\left( {\frac{\partial P}{\partial V}} \right) _{S,\zeta }$ and $\left( {\frac{\partial P}{\partial \zeta }} \right) _{S,V} \cdot \left( {\frac{\partial \zeta }{\partial V}} \right) _{S,A}$ are opposite, only mathematics can prove Eq. 91.

$$\begin{aligned} \left( {\frac{\partial A}{\partial \zeta }} \right) _{S,V}> & {} 0 \end{aligned}$$

(90)

$$\begin{aligned} \left| {\left( {\frac{\partial P}{\partial V}} \right) _{S,\zeta } } \right|\ge & {} \left| {\left( {\frac{\partial P}{\partial \zeta }} \right) _{S,V} \cdot \left( {\frac{\partial \zeta }{\partial V}} \right) _{S,A} } \right| \end{aligned}$$

(91)

In fact any pair of variables in a thermodynamic function, such as S and T, P and V, $\upmu _{\mathrm{i}}$ and $\hbox {n}_{\mathrm{i}}$, and A and $\upzeta$ in Eq. 87 for internal energy U or for other thermodynamic functions as enthalpy H, Helmholtz free energy F, Gibbs free energy G are conjugated variables.

1.4 Appendix 4: Symbols used

• P pressure of the system.

• V volume of system.

• T temperature of the system.

• q heat absorbed in the system.

• w in Eq. 1 is the initial mole number of $\hbox {N}_{2}$.

• t time.

• R in Eq. 19 is the universal gas constant.

• $\hbox {A}_{\mathrm{i}}$ in Eq. 2 is the chemical formula for species i in chemical reaction.

• i and j indices for species in a system. r and p are used for reactant and product respectively . N is the total number of species in the reacting system. $\hbox {N}_{\mathrm{r}}$ and $\hbox {N}_{\mathrm{p}}$ are the total number of reactants and products, respectively.

• $\upzeta$ the reaction extent. It is defined as $\zeta =\frac{n_i -n_i^0 }{v_i }$ along with Eq. 1.

• $v_i$ the coefficient for species i in a balanced chemical reaction. Its value is positive for product and negative for reactant

• $\Delta v$ the sum of the coefficients for all species in a balanced chemical reaction.

  $$\begin{aligned} \Delta v=\sum _i {v_i } =\sum _p {v_p } -\sum _r {\left| {v_r}\right| }. \end{aligned}$$

• $\hbox {n}_{\mathrm{i}}$ the amount of species i while $n_i^0$ is for initial mole number. $n_i =n_i^0 +v_i \xi$.

• $\hbox {n}_{\mathrm{T}}$ total amount of all species in chemical reacting system. $n_T =\sum \limits _i {n_i } =\Delta v\zeta +\sum \limits _i{n_i^0}$.

• $\hbox {x}_{\mathrm{i}}$ mole fraction for species i. $x_i =\frac{n_i}{n_T }$.

• $\hbox {Q}_{\mathrm{x}}$ reaction quotient expressed in mole fractions. $Q_x =\prod \nolimits _i {x_i^{v_i } } =n_T^{-\Delta v} \prod \nolimits _i {n_i^{v_i } }$. It is a unitless quantity. At equilibrium $\hbox {Q}_{\mathrm{x}}=\hbox {K}_{\mathrm{x}}$. When the reaction quotient is expressed in partial pressure, $\hbox {Q}_{\mathrm{p}}$ is used and the corresponding equilibrium constant is $\hbox {K}_{\mathrm{p}}$. $\hbox {Q}_{\mathrm{c}}$ is expressed in molarities as shown in Eq. 21. $\hbox {N}_{\mathrm{x}}$ and $\hbox {D}_{\mathrm{x}}$ are related to $\hbox {Q}_{\mathrm{x}}$ and defined as $N_x =\prod \nolimits _i {n_i^{v_i}},\,D_x =\left( \frac{1}{n_T}\right) ^{\Delta v}$.

• $\hbox {K}_{\mathrm{x}}$ an equilibrium constant formulated from mole fractions for gaseous chemical reaction at constant T and P.

• $\hbox {K}_{\mathrm{p}}$ an equilibrium constant formulated from partial pressures for chemical reactions at constant T.

• G Gibbs energy.

• $\mu _j$ the chemical potential for species j. $\mu _j^0 (T)$ is the standard chemical potential at temperature T.