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

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

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111981: 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	0000111981: 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	0000111975: 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	0000111556: 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}.	9781951738 $\kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T$ 3497828859 $V = \frac{n R T}{P}$		8368984890 $\kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T$	LHS diff is -pdg0004645 + pdg0007586 RHS diff is pdg0002834pdg0007343pdg0008179/pdg0008134 - pdg0002834pdg0007343pdg0008179/(pdg0007586pdg0008134*2)
5	0000111457: 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 $\kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T$	valid
6	0000111457: 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 $\kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right)$	valid
7	0000111556: 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	0000111457: 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	0000111341: 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

Symbols used in coefficient of isothermal compressibility using the equation of state for an ideal gas

0000004645: $ \kappa_T $
0000002834: $ n $
0000008134: $ P $
0000008179: $ R $
0000007343: $ T $
0000007586: $ V $

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0.005 seconds for pdg_app/to_review_derivation 4d1853a8-2c47-485f-9ac6-490a109b83d8
0.006 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivationcf807eb5-deae-42f5-ad4f-4fc99456926e
0.005 seconds for compute/input_feed_output_infrule_for_step: get_inference_rule_connected_to_step_IDcf807eb5-deae-42f5-ad4f-4fc99456926e
0.005 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type HAS_INPUTcf807eb5-deae-42f5-ad4f-4fc99456926e
0.005 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_FEEDcf807eb5-deae-42f5-ad4f-4fc99456926e
0.005 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_OUTPUTcf807eb5-deae-42f5-ad4f-4fc99456926e
0.005 seconds for compute/get_dict_of_steps_in_derivation: get_sequence_index_for_stepcf807eb5-deae-42f5-ad4f-4fc99456926e
0.003 seconds for pdg_app/ 4d1853a8-2c47-485f-9ac6-490a109b83d8

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111981: 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	0000111981: 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	0000111975: 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	0000111556: 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}.	9781951738 \(\kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T\) 3497828859 \(V = \frac{n R T}{P}\)		8368984890 \(\kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T\)	LHS diff is -pdg0004645 + pdg0007586 RHS diff is pdg0002834pdg0007343pdg0008179/pdg0008134 - pdg0002834pdg0007343pdg0008179/(pdg0007586pdg0008134*2)
5	0000111457: 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 \(\kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T\)	valid
6	0000111457: 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 \(\kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right)\)	valid
7	0000111556: 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	0000111457: 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	0000111341: 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