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Review coefficient of thermal expansion 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.			3464107376 $\alpha=\frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p$	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.			3497828859 $V=\frac{n R T}{P}$	no validation is available for declarations
3	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}.	3464107376 $\alpha=\frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p$ 3497828859 $V=\frac{n R T}{P}$		1311403394 $\alpha=\frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P$	Type Tuple cannot be instantiated; use tuple() instead
4	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	1311403394 $\alpha=\frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P$		5962145508 $\alpha=\frac{nR}{VP}$	Type Tuple cannot be instantiated; use tuple() instead
5	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
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}.	8435841627 $P V=n R T$	7924842770 $T$	2613006036 $\frac{PV}{T}=nR$	valid
7	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}.	5962145508 $\alpha=\frac{nR}{VP}$ 2613006036 $\frac{PV}{T}=nR$		6925244346 $\alpha=\frac{PV}{T} \frac{1}{VP}$	LHS diff is -pdg0004686 + pdg0007586pdg0008134/pdg0007343 RHS diff is pdg0002834pdg0008179 - pdg0007586*pdg0008134/pdg0007343
8	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	6925244346 $\alpha=\frac{PV}{T} \frac{1}{VP}$		2472653783 $\alpha=\frac{1}{T}$	LHS diff is 0 RHS diff is (pdg0007586*pdg0008134 - 1)/pdg0007343
9	111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	2472653783 $\alpha=\frac{1}{T}$			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.			3464107376 \(\alpha=\frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p\)	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.			3497828859 \(V=\frac{n R T}{P}\)	no validation is available for declarations
3	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}.	3464107376 \(\alpha=\frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p\) 3497828859 \(V=\frac{n R T}{P}\)		1311403394 \(\alpha=\frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P\)	Type Tuple cannot be instantiated; use tuple() instead
4	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	1311403394 \(\alpha=\frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P\)		5962145508 \(\alpha=\frac{nR}{VP}\)	Type Tuple cannot be instantiated; use tuple() instead
5	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
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}.	8435841627 \(P V=n R T\)	7924842770 \(T\)	2613006036 \(\frac{PV}{T}=nR\)	valid
7	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}.	5962145508 \(\alpha=\frac{nR}{VP}\) 2613006036 \(\frac{PV}{T}=nR\)		6925244346 \(\alpha=\frac{PV}{T} \frac{1}{VP}\)	LHS diff is -pdg0004686 + pdg0007586pdg0008134/pdg0007343 RHS diff is pdg0002834pdg0008179 - pdg0007586*pdg0008134/pdg0007343
8	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	6925244346 \(\alpha=\frac{PV}{T} \frac{1}{VP}\)		2472653783 \(\alpha=\frac{1}{T}\)	LHS diff is 0 RHS diff is (pdg0007586*pdg0008134 - 1)/pdg0007343
9	111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	2472653783 \(\alpha=\frac{1}{T}\)			no validation is available for declarations