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

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.
  1. 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.
  1. 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}.
  1. 8435841627
    \(P V = n R T\)
  1. 6296166842
    \(P\)
  1. 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}.
  1. 9781951738
    \(\kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T\)
  2. 3497828859
    \(V = \frac{n R T}{P}\)
  1. 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 pdg0002834*pdg0007343*pdg0008179/pdg0008134 - pdg0002834*pdg0007343*pdg0008179/(pdg0007586*pdg0008134**2)
5
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 8368984890
    \(\kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T\)
  1. 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}.
  1. 1190768176
    \(\kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T\)
  1. 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}.
  1. 3605073197
    \(\kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right)\)
  2. 8435841627
    \(P V = n R T\)
  1. 9847143017
    \(\kappa_T = \frac{-PV}{V} \left( \frac{-1}{P^2}\right)\)
 LHS diff is -pdg0004645 + pdg0007586*pdg0008134 RHS diff is pdg0002834*pdg0007343*pdg0008179 - 1/pdg0008134
8
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 9847143017
    \(\kappa_T = \frac{-PV}{V} \left( \frac{-1}{P^2}\right)\)
  1. 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.
  1. 9718685793
    \(\kappa_T = \frac{1}{P}\)
no validation is available for declarations

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