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Review coefficient of isothermal compressibility using the equation of state for an ideal gas

abstract:
reference: https://notendur.hi.is/hj/EE2/HD1lausn.pdf

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.			9781951738 $\kappa_T=\frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T$	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.			8435841627 $P V=n R T$	no validation is available for declarations
3	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}.	8435841627 $P V=n R T$	6296166842 $P$	3497828859 $V=\frac{n R T}{P}$	valid
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}.	3497828859 $V=\frac{n R T}{P}$ 9781951738 $\kappa_T=\frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T$		8368984890 $\kappa_T=\frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T$	Type Tuple cannot be instantiated; use tuple() instead
5	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	8368984890 $\kappa_T=\frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T$		1190768176 $\kappa_T=\frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T$	Type Tuple cannot be instantiated; use tuple() instead
6	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	1190768176 $\kappa_T=\frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T$		3605073197 $\kappa_T=\frac{-nRT}{V} \left( \frac{-1}{P^2}\right)$	Type Tuple cannot be instantiated; use tuple() instead
7	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}.	3605073197 $\kappa_T=\frac{-nRT}{V} \left( \frac{-1}{P^2}\right)$ 8435841627 $P V=n R T$		9847143017 $\kappa_T=\frac{-PV}{V} \left( \frac{-1}{P^2}\right)$	LHS diff is -pdg0004645 + pdg0007586pdg0008134 RHS diff is pdg0002834pdg0007343*pdg0008179 - 1/pdg0008134
8	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	9847143017 $\kappa_T=\frac{-PV}{V} \left( \frac{-1}{P^2}\right)$		9718685793 $\kappa_T=\frac{1}{P}$	valid
9	111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	9718685793 $\kappa_T=\frac{1}{P}$			no validation is available for declarations

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timing of Neo4j queries:

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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.			9781951738 \(\kappa_T=\frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T\)	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.			8435841627 \(P V=n R T\)	no validation is available for declarations
3	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}.	8435841627 \(P V=n R T\)	6296166842 \(P\)	3497828859 \(V=\frac{n R T}{P}\)	valid
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}.	3497828859 \(V=\frac{n R T}{P}\) 9781951738 \(\kappa_T=\frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T\)		8368984890 \(\kappa_T=\frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T\)	Type Tuple cannot be instantiated; use tuple() instead
5	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	8368984890 \(\kappa_T=\frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T\)		1190768176 \(\kappa_T=\frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T\)	Type Tuple cannot be instantiated; use tuple() instead
6	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	1190768176 \(\kappa_T=\frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T\)		3605073197 \(\kappa_T=\frac{-nRT}{V} \left( \frac{-1}{P^2}\right)\)	Type Tuple cannot be instantiated; use tuple() instead
7	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}.	3605073197 \(\kappa_T=\frac{-nRT}{V} \left( \frac{-1}{P^2}\right)\) 8435841627 \(P V=n R T\)		9847143017 \(\kappa_T=\frac{-PV}{V} \left( \frac{-1}{P^2}\right)\)	LHS diff is -pdg0004645 + pdg0007586pdg0008134 RHS diff is pdg0002834pdg0007343*pdg0008179 - 1/pdg0008134
8	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	9847143017 \(\kappa_T=\frac{-PV}{V} \left( \frac{-1}{P^2}\right)\)		9718685793 \(\kappa_T=\frac{1}{P}\)	valid
9	111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	9718685793 \(\kappa_T=\frac{1}{P}\)			no validation is available for declarations