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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

d3js visualization of steps and expressions in first law of thermodynamics


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