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review derivation: coefficient of thermal expansion using the equation of state for an ideal gas

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Notes for this derivation:
https://notendur.hi.is/hj/EE2/HD1lausn.pdf

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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. 3464107376; locally 5888046:
    \(\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p\)
    \(pdg_{4686} = \frac{\frac{d}{d pdg_{7343}} pdg_{7586}}{pdg_{7586}}\)
no validation is available for declarations 3464107376:
3464107376:
2 declare initial expr
  1. 3497828859; locally 5927974:
    \(V = \frac{n R T}{P}\)
    \(pdg_{7586} = \frac{pdg_{2834} pdg_{7343} pdg_{8179}}{pdg_{8134}}\)
no validation is available for declarations 3497828859:
3497828859:
3 substitute LHS of expr 1 into expr 2
  1. 3497828859; locally 5927974:
    \(V = \frac{n R T}{P}\)
    \(pdg_{7586} = \frac{pdg_{2834} pdg_{7343} pdg_{8179}}{pdg_{8134}}\)
  2. 3464107376; locally 5888046:
    \(\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p\)
    \(pdg_{4686} = \frac{\frac{d}{d pdg_{7343}} pdg_{7586}}{pdg_{7586}}\)
  1. 1311403394; locally 7236464:
    \(\alpha = \frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P\)
    \(pdg_{4686} = \frac{pdg_{2834} pdg_{8179} \frac{d}{d pdg_{7343}} pdg_{7343}}{pdg_{7586} pdg_{8134}}\)
LHS diff is 0 RHS diff is -pdg2834*pdg8179/(pdg7586*pdg8134) + 1/pdg7343 3497828859:
3464107376:
1311403394:
3497828859:
3464107376:
1311403394:
4 simplify
  1. 1311403394; locally 7236464:
    \(\alpha = \frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P\)
    \(pdg_{4686} = \frac{pdg_{2834} pdg_{8179} \frac{d}{d pdg_{7343}} pdg_{7343}}{pdg_{7586} pdg_{8134}}\)
  1. 5962145508; locally 4600503:
    \(\alpha = \frac{nR}{VP}\)
    \(pdg_{4686} = \frac{pdg_{2834} pdg_{8179}}{pdg_{7586} pdg_{8134}}\)
valid 1311403394:
5962145508:
1311403394:
5962145508:
5 declare initial expr
  1. 8435841627; locally 5130250:
    \(P V = n R T\)
    \(pdg_{7586} pdg_{8134} = pdg_{2834} pdg_{7343} pdg_{8179}\)
no validation is available for declarations 8435841627:
8435841627:
6 divide both sides by
  1. 8435841627; locally 5130250:
    \(P V = n R T\)
    \(pdg_{7586} pdg_{8134} = pdg_{2834} pdg_{7343} pdg_{8179}\)
  1. 7924842770:
    \(T\)
    \(pdg_{7343}\)
  1. 2613006036; locally 8283443:
    \(\frac{PV}{T} = nR\)
    \(\frac{pdg_{7586} pdg_{8134}}{pdg_{7343}} = pdg_{2834} pdg_{8179}\)
valid 8435841627:
2613006036:
8435841627:
2613006036:
7 substitute RHS of expr 1 into expr 2
  1. 2613006036; locally 8283443:
    \(\frac{PV}{T} = nR\)
    \(\frac{pdg_{7586} pdg_{8134}}{pdg_{7343}} = pdg_{2834} pdg_{8179}\)
  2. 5962145508; locally 4600503:
    \(\alpha = \frac{nR}{VP}\)
    \(pdg_{4686} = \frac{pdg_{2834} pdg_{8179}}{pdg_{7586} pdg_{8134}}\)
  1. 6925244346; locally 7845152:
    \(\alpha = \frac{PV}{T} \frac{1}{VP}\)
    \(pdg_{4686} = \frac{pdg_{7586} pdg_{8134}}{pdg_{7343}}\)
LHS diff is 0 RHS diff is (-pdg7586*pdg8134 + 1)/pdg7343 2613006036:
5962145508:
6925244346:
2613006036:
5962145508:
6925244346:
8 simplify
  1. 6925244346; locally 7845152:
    \(\alpha = \frac{PV}{T} \frac{1}{VP}\)
    \(pdg_{4686} = \frac{pdg_{7586} pdg_{8134}}{pdg_{7343}}\)
  1. 2472653783; locally 2491768:
    \(\alpha = \frac{1}{T}\)
    \(pdg_{4686} = \frac{1}{pdg_{7343}}\)
LHS diff is 0 RHS diff is (pdg7586*pdg8134 - 1)/pdg7343 6925244346:
2472653783: dimensions are consistent
6925244346:
2472653783: N/A
9 declare final expr
  1. 2472653783; locally 2491768:
    \(\alpha = \frac{1}{T}\)
    \(pdg_{4686} = \frac{1}{pdg_{7343}}\)
no validation is available for declarations 2472653783: dimensions are consistent
2472653783: N/A
Physics Derivation Graph: Steps for coefficient of thermal expansion using the equation of state for an ideal gas

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
8134 variable P
\(P\)
real
  • length: -1
  • time: -2
  • mass: 1
pressure 13
8179 constant R
\(R\)
real
  • length: 2
  • time: -2
  • mass: 1
  • temperature: -1
  • amount of substance: -1
ideal gas constant 8.31446261815324   Jā‹…K^{āˆ’1} mol^{āˆ’1}
8
2834 variable n
\(n\)
real
amount of substance 8
4686 variable \alpha
\(\alpha\)
real
  • temperature: -1
expansion coefficient 6
7586 variable V
\(V\)
real
  • length: 3
volume 15
7343 variable T
\(T\)
real
  • temperature: 1
temperature 18
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