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review derivation: first law of thermodynamics

This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.


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Notes for this derivation:
https://www.youtube.com/watch?v=QTiqF-HtkS0 and https://www.youtube.com/watch?v=3Yls-t3B49U

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
1 declare initial expr
  1. 1815398659; locally 7368252:
    \(U = Q + W\)
    \(pdg_{5786} = pdg_{1088} + pdg_{9432}\)
no validation is available for declarations 1815398659:
1815398659:
2 declare initial expr
  1. 9941599459; locally 2445123:
    \(dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV\)
    \(dU = \frac{d}{d pdg_{7343}} pdg_{5786}\)
no validation is available for declarations 9941599459:
9941599459:
hold volume constant in first term; hold temperature constant in second term
3 declare initial expr
  1. 3547519267; locally 8155541:
    \(S = k_{\rm Boltzmann} \ln \Omega\)
    \(pdg_{1394} = pdg_{1157} \log{\left(pdg_{3434} \right)}\)
no validation is available for declarations 3547519267:
3547519267:
4 declare initial expr
  1. 1085150613; locally 4576755:
    \(C_V = \left(\frac{\partial U}{\partial T}\right)_V\)
    \(pdg_{6682} = \frac{d}{d pdg_{7343}} pdg_{5786}\)
no validation is available for declarations 1085150613:
1085150613:
5 declare initial expr
  1. 5634116660; locally 7384950:
    \(\pi_T = \left(\frac{\partial U}{\partial V}\right)_T\)
    \(pdg_{5480} = \frac{d}{d pdg_{7586}} pdg_{5786}\)
no validation is available for declarations 5634116660:
5634116660:
6 substitute LHS of two expressions into expr
  1. 1085150613; locally 4576755:
    \(C_V = \left(\frac{\partial U}{\partial T}\right)_V\)
    \(pdg_{6682} = \frac{d}{d pdg_{7343}} pdg_{5786}\)
  2. 5634116660; locally 7384950:
    \(\pi_T = \left(\frac{\partial U}{\partial V}\right)_T\)
    \(pdg_{5480} = \frac{d}{d pdg_{7586}} pdg_{5786}\)
  3. 9941599459; locally 2445123:
    \(dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV\)
    \(dU = \frac{d}{d pdg_{7343}} pdg_{5786}\)
  1. 5002539602; locally 5358683:
    \(dU = C_V dT + \pi_T dV\)
    \(dU = dT pdg_{6682} + dV pdg_{5480}\)
failed 1085150613:
5634116660:
9941599459:
5002539602:
1085150613:
5634116660:
9941599459:
5002539602:
7 divide both sides by
  1. 5002539602; locally 5358683:
    \(dU = C_V dT + \pi_T dV\)
    \(dU = dT pdg_{6682} + dV pdg_{5480}\)
  1. 8854422847:
    \(dT\)
    \(pdg_{7343}\)
  1. 6055078815; locally 3830663:
    \(\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p\)
    \(\frac{d}{d pdg_{7343}} pdg_{5786}\)
Nothing to split 5002539602:
6055078815:
5002539602:
6055078815:
8 declare initial expr
  1. 3464107376; locally 2714175:
    \(\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p\)
    \(pdg_{4686} = \frac{\frac{d}{d pdg_{7343}} pdg_{7586}}{pdg_{7586}}\)
no validation is available for declarations 3464107376:
3464107376:
9 multiply both sides by
  1. 3464107376; locally 2714175:
    \(\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p\)
    \(pdg_{4686} = \frac{\frac{d}{d pdg_{7343}} pdg_{7586}}{pdg_{7586}}\)
  1. 5074423401:
    \(V\)
    \(pdg_{7586}\)
  1. 6397683463; locally 7939101:
    \(V \alpha = \left( \frac{\partial V}{\partial T} \right)_p\)
    \(pdg_{4686} pdg_{7586} = \frac{d}{d pdg_{7343}} pdg_{7586}\)
valid 3464107376:
6397683463:
3464107376:
6397683463:
10 substitute LHS of expr 1 into expr 2
  1. 6397683463; locally 7939101:
    \(V \alpha = \left( \frac{\partial V}{\partial T} \right)_p\)
    \(pdg_{4686} pdg_{7586} = \frac{d}{d pdg_{7343}} pdg_{7586}\)
  2. 6055078815; locally 3830663:
    \(\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p\)
    \(\frac{d}{d pdg_{7343}} pdg_{5786}\)
  1. 2257410739; locally 1136968:
    \(\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha\)
    \(\frac{d}{d pdg_{7343}} pdg_{5786}\)
Nothing to split 6397683463:
6055078815:
2257410739:
6397683463:
6055078815:
2257410739:
11 simplify
  1. 2257410739; locally 1136968:
    \(\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha\)
    \(\frac{d}{d pdg_{7343}} pdg_{5786}\)
  1. 7826132469; locally 1189259:
    \(\left(\frac{\partial U}{\partial T}\right)_p = C_V + \pi_T V \alpha\)
    \(\frac{d}{d pdg_{7343}} pdg_{5786}\)
Nothing to split 2257410739:
7826132469:
2257410739:
7826132469:
12 declare initial expr
  1. 9781951738; locally 9670239:
    \(\kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T\)
    \(pdg_{4645} = - \frac{\frac{d}{d pdg_{8134}} pdg_{7586}}{pdg_{7586}}\)
no validation is available for declarations 9781951738: error for dim with 9781951738
9781951738: N/A
Physics Derivation Graph: Steps for first law of thermodynamics

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
4645 variable \kappa_T
\(\kappa_T\)
real
coefficient of isothermal compressibility
  • str_note
6
6682 variable C_V
\(C_V\)
real
  • length: 2
  • time: -2
  • mass: 1
  • temperature: -1
heat capacity at constant volume 2
1157 constant k_{Boltzmann}
\(k_{Boltzmann}\)
['real']
  • length: 2
  • time: -2
  • mass: 1
  • temperature: -1
Boltzmann constant 1.38064852 10^{-23}   meter^2 kilogram second^-2 Kelvin^-1
1
3434 variable \Omega
\(\Omega\)
real
number of microscopic configurations (known as microstates) that are consistent with the macroscopic quantities that characterize the system 1
5480 variable \pi_T
\(\pi_T\)
real
  • length: 1
  • time: -2
  • mass: 1
internal pressure at constant temperature 2
5786 variable U
\(U\)
real
  • length: 2
  • time: -2
  • mass: 1
internal energy 7
8134 variable P
\(P\)
real
  • length: -1
  • time: -2
  • mass: 1
pressure 13
4686 variable \alpha
\(\alpha\)
real
  • temperature: -1
expansion coefficient 6
1394 variable S
\(S\)
real
  • length: 2
  • time: -2
  • mass: 1
  • temperature: -1
entropy 1
1088 variable W
\(W\)
real
  • length: 2
  • time: -2
  • mass: 1
work done to a system 1
7586 variable V
\(V\)
real
  • length: 3
volume 15
7343 variable T
\(T\)
real
  • temperature: 1
temperature 18
9432 variable Q
\(Q\)
real
  • length: 2
  • time: -2
  • mass: 1
heat flow 1
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