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Review first law of thermodynamics

abstract: https://www.youtube.com/watch?v=QTiqF-HtkS0 and https://www.youtube.com/watch?v=3Yls-t3B49U
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.			1815398659 $U=Q + W$	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.			9941599459 $dU=\left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV$	no validation is available for declarations
3	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			3547519267 $S=k_{\rm Boltzmann} \ln \Omega$	no validation is available for declarations
4	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			1085150613 $C_V=\left(\frac{\partial U}{\partial T}\right)_V$	no validation is available for declarations
5	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			5634116660 $\pi_T=\left(\frac{\partial U}{\partial V}\right)_T$	no validation is available for declarations
6	111732: substitute LHS of two expressions into expression number of inputs: 3; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} and LHS of Eq.~\\ref{eq:#2} into Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}.	5634116660 $\pi_T=\left(\frac{\partial U}{\partial V}\right)_T$ 1085150613 $C_V=\left(\frac{\partial U}{\partial T}\right)_V$ 9941599459 $dU=\left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV$		5002539602 $dU=C_V dT + \pi_T dV$	recognized infrule but not yet supported
7	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}.	5002539602 $dU=C_V dT + \pi_T dV$	8854422847 $dT$	6055078815 $\left(\frac{\partial U}{\partial T}\right)_p=C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p$	Type Tuple cannot be instantiated; use tuple() instead
8	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
9	111182: multiply both sides by number of inputs: 1; feeds: 1; outputs: 1 Multiply both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	3464107376 $\alpha=\frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p$	5074423401 $V$	6397683463 $V \alpha=\left( \frac{\partial V}{\partial T} \right)_p$	Type Tuple cannot be instantiated; use tuple() instead
10	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}.	6055078815 $\left(\frac{\partial U}{\partial T}\right)_p=C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p$ 6397683463 $V \alpha=\left( \frac{\partial V}{\partial T} \right)_p$		2257410739 $\left(\frac{\partial U}{\partial T}\right)_p=C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha$	Type Tuple cannot be instantiated; use tuple() instead
11	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	2257410739 $\left(\frac{\partial U}{\partial T}\right)_p=C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha$		7826132469 $\left(\frac{\partial U}{\partial T}\right)_p=C_V + \pi_T V \alpha$	Type Tuple cannot be instantiated; use tuple() instead
12	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

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timing of Neo4j queries:

0.007 seconds for pdg_app/to_review_derivation: node_properties, derivation 1458605
0.004 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivation9228972
0.007 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule1365215
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0.005 seconds for compute/get_dict_of_steps_in_derivation: get_step_has_sequence_index9228972
0.005 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule8719526
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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.			1815398659 \(U=Q + W\)	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.			9941599459 \(dU=\left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV\)	no validation is available for declarations
3	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			3547519267 \(S=k_{\rm Boltzmann} \ln \Omega\)	no validation is available for declarations
4	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			1085150613 \(C_V=\left(\frac{\partial U}{\partial T}\right)_V\)	no validation is available for declarations
5	111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			5634116660 \(\pi_T=\left(\frac{\partial U}{\partial V}\right)_T\)	no validation is available for declarations
6	111732: substitute LHS of two expressions into expression number of inputs: 3; feeds: 0; outputs: 1 Substitute LHS of Eq.~\\ref{eq:#1} and LHS of Eq.~\\ref{eq:#2} into Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}.	5634116660 \(\pi_T=\left(\frac{\partial U}{\partial V}\right)_T\) 1085150613 \(C_V=\left(\frac{\partial U}{\partial T}\right)_V\) 9941599459 \(dU=\left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV\)		5002539602 \(dU=C_V dT + \pi_T dV\)	recognized infrule but not yet supported
7	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}.	5002539602 \(dU=C_V dT + \pi_T dV\)	8854422847 \(dT\)	6055078815 \(\left(\frac{\partial U}{\partial T}\right)_p=C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p\)	Type Tuple cannot be instantiated; use tuple() instead
8	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
9	111182: multiply both sides by number of inputs: 1; feeds: 1; outputs: 1 Multiply both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.	3464107376 \(\alpha=\frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p\)	5074423401 \(V\)	6397683463 \(V \alpha=\left( \frac{\partial V}{\partial T} \right)_p\)	Type Tuple cannot be instantiated; use tuple() instead
10	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}.	6055078815 \(\left(\frac{\partial U}{\partial T}\right)_p=C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p\) 6397683463 \(V \alpha=\left( \frac{\partial V}{\partial T} \right)_p\)		2257410739 \(\left(\frac{\partial U}{\partial T}\right)_p=C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha\)	Type Tuple cannot be instantiated; use tuple() instead
11	111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	2257410739 \(\left(\frac{\partial U}{\partial T}\right)_p=C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha\)		7826132469 \(\left(\frac{\partial U}{\partial T}\right)_p=C_V + \pi_T V \alpha\)	Type Tuple cannot be instantiated; use tuple() instead
12	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