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review derivation: Langmuir Adsorption

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Notes for this derivation:
from https://arxiv.org/pdf/2210.12150.pdf

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
10 divide both sides by
  1. 3488423948; locally 2575148:
    \(k_{\rm adsorption} p_A [S] = k_{\rm desorption} [A_{\rm adsorption}]\)
    \(pdg_{6850} pdg_{9046} pdg_{9067} = pdg_{4940} pdg_{8379}\)
  1. 1945487024:
    \(p_A [S]\)
    \(pdg_{9046} pdg_{9067}\)
  1. 5085809757; locally 2552367:
    \(\frac{k_{\rm adsorption}}{k_{\rm desorption}} = \frac{[A_{\rm adsorption}]}{p_A [S]}\)
    \(\frac{pdg_{6850}}{pdg_{8379}} = \frac{pdg_{4940}}{pdg_{9046} pdg_{9067}}\)
LHS diff is pdg6850 - pdg6850/pdg8379 RHS diff is pdg4940*(pdg8379 - 1)/(pdg9046*pdg9067) 3488423948:
5085809757:
3488423948:
5085809757:
11 declare initial expr
  1. 7924063906; locally 5044727:
    \(K_{equilibrium} = \frac{k_{\rm adsorption}}{k_{\rm desorption}}\)
    \(pdg_{4933} = \frac{pdg_{6850}}{pdg_{8379}}\)
no validation is available for declarations 7924063906:
7924063906:
definition of equilibrium
6 divide both sides by
  1. 3488423948; locally 2575148:
    \(k_{\rm adsorption} p_A [S] = k_{\rm desorption} [A_{\rm adsorption}]\)
    \(pdg_{6850} pdg_{9046} pdg_{9067} = pdg_{4940} pdg_{8379}\)
  1. 8162179726:
    \(k_{\rm adsorption} p_A\)
    \(pdg_{6850} pdg_{9046}\)
  1. 9562264720; locally 5003983:
    \([S] = \frac{k_{\rm desorption} [A_{\rm adsorption}]}{k_{\rm adsorption} p_A}\)
    \(pdg_{9067} = \frac{pdg_{4940} pdg_{8379}}{pdg_{6850} pdg_{9046}}\)
valid 3488423948:
9562264720:
3488423948:
9562264720:
16 raise both sides to power
  1. 2114909846; locally 7042641:
    \(\theta_A = \frac{[A_{\rm adsorption}]}{[S_0]}\)
    \(pdg_{1791} = \frac{pdg_{4940}}{pdg_{3037}}\)
  1. 7564010952:
    \(-1\)
    \(-1\)
  1. 8131665171; locally 4039036:
    \(\frac{1}{\theta_A} = \frac{[S_0]}{[A_{\rm adsorption}]}\)
    \(\frac{1}{pdg_{1791}} = \frac{pdg_{3037}}{pdg_{4940}}\)
no check is performed 2114909846:
8131665171:
2114909846:
8131665171:
12 raise both sides to power
  1. 7924063906; locally 5044727:
    \(K_{equilibrium} = \frac{k_{\rm adsorption}}{k_{\rm desorption}}\)
    \(pdg_{4933} = \frac{pdg_{6850}}{pdg_{8379}}\)
  1. 5516739892:
    \(-1\)
    \(-1\)
  1. 6240546932; locally 8620451:
    \(\frac{1}{K_{equilibrium}} = \frac{k_{\rm desorption}}{k_{\rm adsorption}}\)
    \(\frac{1}{pdg_{4933}} = \frac{pdg_{8379}}{pdg_{6850}}\)
no check is performed 7924063906:
6240546932:
7924063906:
6240546932:
18 change variable X to Y
  1. 7928111771; locally 8754546:
    \(\frac{1}{\theta_A} = \frac{1}{K_{\rm equilibrium} p_A} + 1\)
    \(\frac{1}{pdg_{1791}} = 1 + \frac{1}{pdg_{4933} pdg_{9046}}\)
  1. 6346902704:
    \(1\)
    \(1\)
  2. 7630953440:
    \(\frac{K_{\rm equilibrium} p_A}{K_{\rm equilibrium} p_A}\)
    \(1\)
  1. 7267424860; locally 4829867:
    \(\frac{1}{\theta_A} = \frac{1+(K_{\rm equilibrium}\ p_A)}{K_{\rm equilibrium}\ p_A}\)
    \(\frac{1}{pdg_{1791}} = \frac{pdg_{4933} pdg_{9046} + 1}{pdg_{4933} pdg_{9046}}\)
valid 7928111771:
7267424860:
7928111771:
7267424860:
14 divide both sides by
  1. 7517073655; locally 4487508:
    \([S_0] = \left(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]\)
    \(pdg_{3037} = pdg_{4940} \left(1 + \frac{1}{pdg_{4933} pdg_{9046}}\right)\)
  1. 5426418187:
    \([A_{\rm adsorption}]\)
    \(pdg_{4940}\)
  1. 6457999644; locally 5382248:
    \(\frac{[S_0]}{[A_{\rm adsorption}]} = \frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\)
    \(\frac{pdg_{3037}}{pdg_{4940}} = 1 + \frac{1}{pdg_{4933} pdg_{9046}}\)
valid 7517073655:
6457999644:
7517073655:
6457999644:
17 substitute RHS of expr 1 into expr 2
  1. 8131665171; locally 4039036:
    \(\frac{1}{\theta_A} = \frac{[S_0]}{[A_{\rm adsorption}]}\)
    \(\frac{1}{pdg_{1791}} = \frac{pdg_{3037}}{pdg_{4940}}\)
  2. 6457999644; locally 5382248:
    \(\frac{[S_0]}{[A_{\rm adsorption}]} = \frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\)
    \(\frac{pdg_{3037}}{pdg_{4940}} = 1 + \frac{1}{pdg_{4933} pdg_{9046}}\)
  1. 7928111771; locally 8754546:
    \(\frac{1}{\theta_A} = \frac{1}{K_{\rm equilibrium} p_A} + 1\)
    \(\frac{1}{pdg_{1791}} = 1 + \frac{1}{pdg_{4933} pdg_{9046}}\)
valid 8131665171:
6457999644:
7928111771:
8131665171:
6457999644:
7928111771:
19 raise both sides to power
  1. 7267424860; locally 4829867:
    \(\frac{1}{\theta_A} = \frac{1+(K_{\rm equilibrium}\ p_A)}{K_{\rm equilibrium}\ p_A}\)
    \(\frac{1}{pdg_{1791}} = \frac{pdg_{4933} pdg_{9046} + 1}{pdg_{4933} pdg_{9046}}\)
  1. 3911081515:
    \(-1\)
    \(-1\)
  1. 4689334676; locally 8722827:
    \(\theta_A = \frac{K_{\rm equilibrium}\ p_A}{1+K_{\rm equilibrium}\ p_A}\)
    \(pdg_{1791} = \frac{pdg_{4933} pdg_{9046}}{pdg_{4933} pdg_{9046} + 1}\)
no check is performed 7267424860:
4689334676:
7267424860:
4689334676:
3 declare assumption
  1. 6783009163; locally 9492936:
    \(r_{\rm adsorption} = r_{\rm desorption}\)
    \(pdg_{6687} = pdg_{1966}\)
no validation is available for declarations 6783009163:
6783009163:
5 substitute LHS of expr 1 into expr 2
  1. 6955192897; locally 8859060:
    \(r_{\rm desorption} = k_{\rm desorption} [A_{\rm adsorption}]\)
    \(pdg_{1966} = pdg_{4940} pdg_{8379}\)
  2. 3507029294; locally 1615903:
    \(k_{\rm adsorption} p_A [S] = r_{\rm desorption}\)
    \(pdg_{6850} pdg_{9046} pdg_{9067} = pdg_{1966}\)
  1. 3488423948; locally 2575148:
    \(k_{\rm adsorption} p_A [S] = k_{\rm desorption} [A_{\rm adsorption}]\)
    \(pdg_{6850} pdg_{9046} pdg_{9067} = pdg_{4940} pdg_{8379}\)
valid 6955192897:
3507029294:
3488423948:
6955192897:
3507029294:
3488423948:
15 declare initial expr
  1. 2114909846; locally 7042641:
    \(\theta_A = \frac{[A_{\rm adsorption}]}{[S_0]}\)
    \(pdg_{1791} = \frac{pdg_{4940}}{pdg_{3037}}\)
no validation is available for declarations 2114909846:
2114909846:
4 substitute LHS of expr 1 into expr 2
  1. 3736177473; locally 7133735:
    \(r_{\rm adsorption} = k_{\rm adsorption} p_A [S]\)
    \(pdg_{6687} = pdg_{6850} pdg_{9046} pdg_{9067}\)
  2. 6783009163; locally 9492936:
    \(r_{\rm adsorption} = r_{\rm desorption}\)
    \(pdg_{6687} = pdg_{1966}\)
  1. 3507029294; locally 1615903:
    \(k_{\rm adsorption} p_A [S] = r_{\rm desorption}\)
    \(pdg_{6850} pdg_{9046} pdg_{9067} = pdg_{1966}\)
valid 3736177473: inconsistent dimensions
6783009163:
3507029294:
3736177473: N/A
6783009163:
3507029294:
2 declare initial expr
  1. 6955192897; locally 8859060:
    \(r_{\rm desorption} = k_{\rm desorption} [A_{\rm adsorption}]\)
    \(pdg_{1966} = pdg_{4940} pdg_{8379}\)
no validation is available for declarations 6955192897:
6955192897:
20 declare final expr
  1. 4689334676; locally 8722827:
    \(\theta_A = \frac{K_{\rm equilibrium}\ p_A}{1+K_{\rm equilibrium}\ p_A}\)
    \(pdg_{1791} = \frac{pdg_{4933} pdg_{9046}}{pdg_{4933} pdg_{9046} + 1}\)
no validation is available for declarations 4689334676:
4689334676:
7 declare initial expr
  1. 3599953931; locally 7265984:
    \([S_0] = [S] + [A_{\rm adsorption}]\)
    \(pdg_{3037} = pdg_{4940} + pdg_{9067}\)
no validation is available for declarations 3599953931:
3599953931:
The concentration of all sites by summing the concentration of free sites [S] and occupied sites
8 substitute LHS of expr 1 into expr 2
  1. 9562264720; locally 5003983:
    \([S] = \frac{k_{\rm desorption} [A_{\rm adsorption}]}{k_{\rm adsorption} p_A}\)
    \(pdg_{9067} = \frac{pdg_{4940} pdg_{8379}}{pdg_{6850} pdg_{9046}}\)
  2. 3599953931; locally 7265984:
    \([S_0] = [S] + [A_{\rm adsorption}]\)
    \(pdg_{3037} = pdg_{4940} + pdg_{9067}\)
  1. 4301729661; locally 1337055:
    \([S_0] = \frac{[A_{\rm adsorption}]}{\left( \frac{k_{\rm adsorption}}{k_{\rm desorption}} \right) p_A} + [A_{\rm adsorption}]\)
    \(pdg_{3037} = pdg_{4940} + \frac{pdg_{4940} pdg_{8379}}{pdg_{6850} pdg_{9046}}\)
valid 9562264720:
3599953931:
4301729661:
9562264720:
3599953931:
4301729661:
9 factor out X
  1. 4301729661; locally 1337055:
    \([S_0] = \frac{[A_{\rm adsorption}]}{\left( \frac{k_{\rm adsorption}}{k_{\rm desorption}} \right) p_A} + [A_{\rm adsorption}]\)
    \(pdg_{3037} = pdg_{4940} + \frac{pdg_{4940} pdg_{8379}}{pdg_{6850} pdg_{9046}}\)
  1. 1268845856:
    \([A_{\rm adsorption}]\)
    \(pdg_{4940}\)
  1. 2168306601; locally 9195751:
    \([S_0] = \left(\frac{k_{\rm desorption}}{k_{\rm adsorption}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]\)
    \(pdg_{3037} = pdg_{4940} \left(1 + \frac{pdg_{8379}}{pdg_{6850} pdg_{9046}}\right)\)
valid 4301729661:
2168306601:
4301729661:
2168306601:
13 substitute RHS of expr 1 into expr 2
  1. 6240546932; locally 8620451:
    \(\frac{1}{K_{equilibrium}} = \frac{k_{\rm desorption}}{k_{\rm adsorption}}\)
    \(\frac{1}{pdg_{4933}} = \frac{pdg_{8379}}{pdg_{6850}}\)
  2. 2168306601; locally 9195751:
    \([S_0] = \left(\frac{k_{\rm desorption}}{k_{\rm adsorption}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]\)
    \(pdg_{3037} = pdg_{4940} \left(1 + \frac{pdg_{8379}}{pdg_{6850} pdg_{9046}}\right)\)
  1. 7517073655; locally 4487508:
    \([S_0] = \left(\frac{1}{K_{\rm equilibrium}} \frac{1}{p_A} + 1\right)[A_{\rm adsorption}]\)
    \(pdg_{3037} = pdg_{4940} \left(1 + \frac{1}{pdg_{4933} pdg_{9046}}\right)\)
valid 6240546932:
2168306601:
7517073655:
6240546932:
2168306601:
7517073655:
1 declare initial expr
  1. 3736177473; locally 7133735:
    \(r_{\rm adsorption} = k_{\rm adsorption} p_A [S]\)
    \(pdg_{6687} = pdg_{6850} pdg_{9046} pdg_{9067}\)
no validation is available for declarations 3736177473: inconsistent dimensions
3736177473: N/A
Physics Derivation Graph: Steps for Langmuir Adsorption

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
6850 variable k_{\rm adsorption}
\(k_{\rm adsorption}\)
real dimensionless constant of forward adsorption reaction 10
9046 variable p_A
\(p_A\)
real dimensionless partial pressure of A over the surface 15
3037 variable [S_0]
\([S_0]\)
real
  • amount of substance: 1
total number of sites 7
1966 variable r_{\rm desorption}
\(r_{\rm desorption}\)
real
  • time: -1
rate of desorption 3
9067 variable [S]
\([S]\)
real
  • amount of substance: 1
  • length: -2
concentration of free sites in number per square meter 7
4940 variable [A_{\rm adsorption}]
\([A_{\rm adsorption}]\)
real
  • amount of substance: 1
  • length: 2
surface concentration of A in molecules per square meter 13
8379 variable k_{\rm desorption}
\(k_{\rm desorption}\)
real dimensionless constant of backward desorption reaction 8
4933 variable K_{\rm equilibrium}
\(K_{\rm equilibrium}\)
real dimensionless constant for equilibrium when rate of adsorption equals the rate of desorption 8
6687 variable r_{\rm adsorption}
\(r_{\rm adsorption}\)
real
  • time: -1
rate of adsorption 2
1791 variable \theta_A
\(\theta_A\)
real dimensionless the fraction of the surface sites covered with A 5
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