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Review Langmuir Adsorption

abstract: from https://arxiv.org/pdf/2210.12150.pdf
reference:

step	inference rule	input	feed	output	step validity (as per SymPy)
1	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			3736177473 $r_{\rm adsorption}=k_{\rm adsorption} p_A [S]$	no validation is available for declarations
2	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			6955192897 $r_{\rm desorption}=k_{\rm desorption} [A_{\rm adsorption}]$	no validation is available for declarations
3	111104: declare assumption number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an assumption.			6783009163 $r_{\rm adsorption}=r_{\rm desorption}$	no validation is available for declarations
4	111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	3736177473 $r_{\rm adsorption}=k_{\rm adsorption} p_A [S]$ 6783009163 $r_{\rm adsorption}=r_{\rm desorption}$		3507029294 $k_{\rm adsorption} p_A [S]=r_{\rm desorption}$	valid
5	111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	3507029294 $k_{\rm adsorption} p_A [S]=r_{\rm desorption}$ 6955192897 $r_{\rm desorption}=k_{\rm desorption} [A_{\rm adsorption}]$		3488423948 $k_{\rm adsorption} p_A [S]=k_{\rm desorption} [A_{\rm adsorption}]$	LHS diff is pdg0001966 - pdg0006850pdg0009046pdg0009067 RHS diff is 0
6	111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	3488423948 $k_{\rm adsorption} p_A [S]=k_{\rm desorption} [A_{\rm adsorption}]$	8162179726 $k_{\rm adsorption} p_A$	9562264720 $[S]=\frac{k_{\rm desorption} [A_{\rm adsorption}]}{k_{\rm adsorption} p_A}$	valid
7	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			3599953931 $[S_0]=[S] + [A_{\rm adsorption}]$	no validation is available for declarations
8	111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	9562264720 $[S]=\frac{k_{\rm desorption} [A_{\rm adsorption}]}{k_{\rm adsorption} p_A}$ 3599953931 $[S_0]=[S] + [A_{\rm adsorption}]$		4301729661 $[S_0]=\frac{[A_{\rm adsorption}]}{\left( \frac{k_{\rm adsorption}}{k_{\rm desorption}} \right) p_A} + [A_{\rm adsorption}]$	valid
9	111432: factor out X number of inputs: 1; feeds: 1; outputs: 1 Factor $#1$ from Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	4301729661 $[S_0]=\frac{[A_{\rm adsorption}]}{\left( \frac{k_{\rm adsorption}}{k_{\rm desorption}} \right) p_A} + [A_{\rm adsorption}]$	1268845856 $[A_{\rm adsorption}]$	2168306601 $[S_0]=\left(\frac{k_{\rm desorption}}{k_{\rm adsorption}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]$	valid
10	111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	3488423948 $k_{\rm adsorption} p_A [S]=k_{\rm desorption} [A_{\rm adsorption}]$	1945487024 $p_A [S]$	5085809757 $\frac{k_{\rm adsorption}}{k_{\rm desorption}}=\frac{[A_{\rm adsorption}]}{p_A [S]}$	Algebraic error: LHS diff is pdg0006850 - pdg0006850/pdg0008379, RHS diff is pdg0004940(pdg0008379 - 1)/(pdg0009046pdg0009067)
11	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			7924063906 $K_{equilibrium}=\frac{k_{\rm adsorption}}{k_{\rm desorption}}$	no validation is available for declarations
12	111483: raise both sides to power number of inputs: 1; feeds: 1; outputs: 1 Raise both sides of Eq.~\\ref{eq:#2} to $#1$; yields Eq.~\\ref{eq:#3}.	7924063906 $K_{equilibrium}=\frac{k_{\rm adsorption}}{k_{\rm desorption}}$	5516739892 $-1$	6240546932 $\frac{1}{K_{equilibrium}}=\frac{k_{\rm desorption}}{k_{\rm adsorption}}$	recognized infrule but not yet supported
13	111634: substitute RHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute RHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	6240546932 $\frac{1}{K_{equilibrium}}=\frac{k_{\rm desorption}}{k_{\rm adsorption}}$ 2168306601 $[S_0]=\left(\frac{k_{\rm desorption}}{k_{\rm adsorption}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]$		7517073655 $[S_0]=\left(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]$	valid
14	111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	7517073655 $[S_0]=\left(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]$	5426418187 $[A_{\rm adsorption}]$	6457999644 $\frac{[S_0]}{[A_{\rm adsorption}]}=\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1$	valid
15	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			2114909846 $\theta_A=\frac{[A_{\rm adsorption}]}{[S_0]}$	no validation is available for declarations
16	111483: raise both sides to power number of inputs: 1; feeds: 1; outputs: 1 Raise both sides of Eq.~\\ref{eq:#2} to $#1$; yields Eq.~\\ref{eq:#3}.	2114909846 $\theta_A=\frac{[A_{\rm adsorption}]}{[S_0]}$	7564010952 $-1$	8131665171 $\frac{1}{\theta_A}=\frac{[S_0]}{[A_{\rm adsorption}]}$	recognized infrule but not yet supported
17	111634: substitute RHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute RHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	6457999644 $\frac{[S_0]}{[A_{\rm adsorption}]}=\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1$ 8131665171 $\frac{1}{\theta_A}=\frac{[S_0]}{[A_{\rm adsorption}]}$		7928111771 $\frac{1}{\theta_A}=\frac{1}{K_{\rm equilibrium} p_A} + 1$	LHS diff is 0 RHS diff is pdg0003037/pdg0004940 - 1 - 1/(pdg0004933*pdg0009046)
18	111886: change variable X to Y number of inputs: 1; feeds: 2; outputs: 1 Change variable $#1$ to $#2$ in Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}.	7928111771 $\frac{1}{\theta_A}=\frac{1}{K_{\rm equilibrium} p_A} + 1$	7630953440 $\frac{K_{\rm equilibrium} p_A}{K_{\rm equilibrium} p_A}$ 6346902704 $1$	7267424860 $\frac{1}{\theta_A}=\frac{1+(K_{\rm equilibrium}\ p_A)}{K_{\rm equilibrium}\ p_A}$	valid
19	111483: raise both sides to power number of inputs: 1; feeds: 1; outputs: 1 Raise both sides of Eq.~\\ref{eq:#2} to $#1$; yields Eq.~\\ref{eq:#3}.	7267424860 $\frac{1}{\theta_A}=\frac{1+(K_{\rm equilibrium}\ p_A)}{K_{\rm equilibrium}\ p_A}$	3911081515 $-1$	4689334676 $\theta_A=\frac{K_{\rm equilibrium}\ p_A}{1+K_{\rm equilibrium}\ p_A}$	recognized infrule but not yet supported
20	111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	4689334676 $\theta_A=\frac{K_{\rm equilibrium}\ p_A}{1+K_{\rm equilibrium}\ p_A}$			no validation is available for declarations

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step	inference rule	input	feed	output	step validity (as per SymPy)
1	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			3736177473 \(r_{\rm adsorption}=k_{\rm adsorption} p_A [S]\)	no validation is available for declarations
2	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			6955192897 \(r_{\rm desorption}=k_{\rm desorption} [A_{\rm adsorption}]\)	no validation is available for declarations
3	111104: declare assumption number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an assumption.			6783009163 \(r_{\rm adsorption}=r_{\rm desorption}\)	no validation is available for declarations
4	111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	3736177473 \(r_{\rm adsorption}=k_{\rm adsorption} p_A [S]\) 6783009163 \(r_{\rm adsorption}=r_{\rm desorption}\)		3507029294 \(k_{\rm adsorption} p_A [S]=r_{\rm desorption}\)	valid
5	111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	3507029294 \(k_{\rm adsorption} p_A [S]=r_{\rm desorption}\) 6955192897 \(r_{\rm desorption}=k_{\rm desorption} [A_{\rm adsorption}]\)		3488423948 \(k_{\rm adsorption} p_A [S]=k_{\rm desorption} [A_{\rm adsorption}]\)	LHS diff is pdg0001966 - pdg0006850pdg0009046pdg0009067 RHS diff is 0
6	111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	3488423948 \(k_{\rm adsorption} p_A [S]=k_{\rm desorption} [A_{\rm adsorption}]\)	8162179726 \(k_{\rm adsorption} p_A\)	9562264720 \([S]=\frac{k_{\rm desorption} [A_{\rm adsorption}]}{k_{\rm adsorption} p_A}\)	valid
7	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			3599953931 \([S_0]=[S] + [A_{\rm adsorption}]\)	no validation is available for declarations
8	111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	9562264720 \([S]=\frac{k_{\rm desorption} [A_{\rm adsorption}]}{k_{\rm adsorption} p_A}\) 3599953931 \([S_0]=[S] + [A_{\rm adsorption}]\)		4301729661 \([S_0]=\frac{[A_{\rm adsorption}]}{\left( \frac{k_{\rm adsorption}}{k_{\rm desorption}} \right) p_A} + [A_{\rm adsorption}]\)	valid
9	111432: factor out X number of inputs: 1; feeds: 1; outputs: 1 Factor $#1$ from Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	4301729661 \([S_0]=\frac{[A_{\rm adsorption}]}{\left( \frac{k_{\rm adsorption}}{k_{\rm desorption}} \right) p_A} + [A_{\rm adsorption}]\)	1268845856 \([A_{\rm adsorption}]\)	2168306601 \([S_0]=\left(\frac{k_{\rm desorption}}{k_{\rm adsorption}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]\)	valid
10	111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	3488423948 \(k_{\rm adsorption} p_A [S]=k_{\rm desorption} [A_{\rm adsorption}]\)	1945487024 \(p_A [S]\)	5085809757 \(\frac{k_{\rm adsorption}}{k_{\rm desorption}}=\frac{[A_{\rm adsorption}]}{p_A [S]}\)	Algebraic error: LHS diff is pdg0006850 - pdg0006850/pdg0008379, RHS diff is pdg0004940(pdg0008379 - 1)/(pdg0009046pdg0009067)
11	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			7924063906 \(K_{equilibrium}=\frac{k_{\rm adsorption}}{k_{\rm desorption}}\)	no validation is available for declarations
12	111483: raise both sides to power number of inputs: 1; feeds: 1; outputs: 1 Raise both sides of Eq.~\\ref{eq:#2} to $#1$; yields Eq.~\\ref{eq:#3}.	7924063906 \(K_{equilibrium}=\frac{k_{\rm adsorption}}{k_{\rm desorption}}\)	5516739892 \(-1\)	6240546932 \(\frac{1}{K_{equilibrium}}=\frac{k_{\rm desorption}}{k_{\rm adsorption}}\)	recognized infrule but not yet supported
13	111634: substitute RHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute RHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	6240546932 \(\frac{1}{K_{equilibrium}}=\frac{k_{\rm desorption}}{k_{\rm adsorption}}\) 2168306601 \([S_0]=\left(\frac{k_{\rm desorption}}{k_{\rm adsorption}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]\)		7517073655 \([S_0]=\left(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]\)	valid
14	111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	7517073655 \([S_0]=\left(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]\)	5426418187 \([A_{\rm adsorption}]\)	6457999644 \(\frac{[S_0]}{[A_{\rm adsorption}]}=\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\)	valid
15	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			2114909846 \(\theta_A=\frac{[A_{\rm adsorption}]}{[S_0]}\)	no validation is available for declarations
16	111483: raise both sides to power number of inputs: 1; feeds: 1; outputs: 1 Raise both sides of Eq.~\\ref{eq:#2} to $#1$; yields Eq.~\\ref{eq:#3}.	2114909846 \(\theta_A=\frac{[A_{\rm adsorption}]}{[S_0]}\)	7564010952 \(-1\)	8131665171 \(\frac{1}{\theta_A}=\frac{[S_0]}{[A_{\rm adsorption}]}\)	recognized infrule but not yet supported
17	111634: substitute RHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute RHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.	6457999644 \(\frac{[S_0]}{[A_{\rm adsorption}]}=\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\) 8131665171 \(\frac{1}{\theta_A}=\frac{[S_0]}{[A_{\rm adsorption}]}\)		7928111771 \(\frac{1}{\theta_A}=\frac{1}{K_{\rm equilibrium} p_A} + 1\)	LHS diff is 0 RHS diff is pdg0003037/pdg0004940 - 1 - 1/(pdg0004933*pdg0009046)
18	111886: change variable X to Y number of inputs: 1; feeds: 2; outputs: 1 Change variable $#1$ to $#2$ in Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}.	7928111771 \(\frac{1}{\theta_A}=\frac{1}{K_{\rm equilibrium} p_A} + 1\)	7630953440 \(\frac{K_{\rm equilibrium} p_A}{K_{\rm equilibrium} p_A}\) 6346902704 \(1\)	7267424860 \(\frac{1}{\theta_A}=\frac{1+(K_{\rm equilibrium}\ p_A)}{K_{\rm equilibrium}\ p_A}\)	valid
19	111483: raise both sides to power number of inputs: 1; feeds: 1; outputs: 1 Raise both sides of Eq.~\\ref{eq:#2} to $#1$; yields Eq.~\\ref{eq:#3}.	7267424860 \(\frac{1}{\theta_A}=\frac{1+(K_{\rm equilibrium}\ p_A)}{K_{\rm equilibrium}\ p_A}\)	3911081515 \(-1\)	4689334676 \(\theta_A=\frac{K_{\rm equilibrium}\ p_A}{1+K_{\rm equilibrium}\ p_A}\)	recognized infrule but not yet supported
20	111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	4689334676 \(\theta_A=\frac{K_{\rm equilibrium}\ p_A}{1+K_{\rm equilibrium}\ p_A}\)			no validation is available for declarations