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Review first law of thermodynamics

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. 1815398659
    \(U = Q + W\)
 no validation is available for declarations
Note about step 2: hold volume constant in first term; hold temperature constant in second term
2
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 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
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 3547519267
    \(S = k_{\rm Boltzmann} \ln \Omega\)
 no validation is available for declarations
4
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 1085150613
    \(C_V = \left(\frac{\partial U}{\partial T}\right)_V\)
 no validation is available for declarations
5
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 5634116660
    \(\pi_T = \left(\frac{\partial U}{\partial V}\right)_T\)
 no validation is available for declarations
6
  • 0000111732: 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}.
  1. 9941599459
    \(dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV\)
  2. 5634116660
    \(\pi_T = \left(\frac{\partial U}{\partial V}\right)_T\)
  3. 1085150613
    \(C_V = \left(\frac{\partial U}{\partial T}\right)_V\)
  1. 5002539602
    \(dU = C_V dT + \pi_T dV\)
 recognized infrule but not yet supported
7
  • 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. 5002539602
    \(dU = C_V dT + \pi_T dV\)
  1. 8854422847
    \(dT\)
  1. 6055078815
    \(\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p\)
 Not evaluated due to missing term in SymPy
8
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 3464107376
    \(\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p\)
 no validation is available for declarations
9
  • 0000111182: 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}.
  1. 3464107376
    \(\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p\)
  1. 5074423401
    \(V\)
  1. 6397683463
    \(V \alpha = \left( \frac{\partial V}{\partial T} \right)_p\)
 valid
10
  • 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. 6055078815
    \(\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p\)
  2. 6397683463
    \(V \alpha = \left( \frac{\partial V}{\partial T} \right)_p\)
  1. 2257410739
    \(\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha\)
 Not evaluated due to missing term in SymPy
11
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 2257410739
    \(\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha\)
  1. 7826132469
    \(\left(\frac{\partial U}{\partial T}\right)_p = C_V + \pi_T V \alpha\)
 Not evaluated due to missing term in SymPy
12
  • 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

Symbols used in first law of thermodynamics

Steps and expressions for first law of thermodynamics

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