View source for Discussion: Estimation of Specific Heat for Mineral Species

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'''Related Links:''' [[Specific Heat]]
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== Heat Capacity Estimation ==
[[File:dulongp01.png|link=|right]] Here's a fun fact! The molar heat capacity for many solid elements is about 25 J/mol.K. For reference, you can see a SysCAD summary of common elements on the right.

In this discussion page we will look at why this is so, how we can use this property to check if specific heat values are reasonable and estimate the specific heat of minerals for which we have no data.

=== Dulong-Petit Law ===

The number 25 turns out to come from the value <math>3\times R</math>, the Universal Gas Constant (~8.3143 J/mol.K). This fundamental constant is available in PGM code (see [[Predefined_Constants_and_Variables#Standard constants|Standard Constant '''Rgc''']]). You have probably used <math>R</math> in an Arrhenius equation or in calculating the ''density'' of an ideal gas:

:<math> \frac\rho M = \frac p{RT} </math> &nbsp; &nbsp; where M is the molecular weight

That many elements share this value is the [https://en.wikipedia.org/wiki/Dulong%E2%80%93Petit_law Dulong-Petit Law]. Below are the molar heat capacities for all elements up to uranium<sup>[1]</sup>:

[[File:GraphHeatCapacityOfTheElementsI2s.png | 700px | link=]]

Apart from halogens (green), alkali metals (dark blue), noble gases (red) and a few other outliers, all elements cluster around the value 25 J/mol.K.

=== Kopp–Neumann Law ===

It turns out we can take this a lot further. [https://en.wikipedia.org/wiki/Kopp%27s_law The Kopp-Neumann Law] (aka Neumann-Kopp Rule (NKR) or simply Kopp's law) states that we can estimate the molar heat capacity of a ''compound'' by adding the molar heat capacities of the individual elements (note that this applies mainly to '''solid''' species). For example, in modelling Bayer liquors and precipitation, we have to account for organic species, which are generally a mixture of formate/acetate/benzoate compounds. Rather than modelling each individual species, we often group these into a ''pseudo-organic'' species with a representative composition determined by analysis. The [[Alumina 3 Bayer Species Model|Bayer3 species model]] includes the psuedo-organic compound Na2C5O7(s), for which we can estimate the molar heat capacity to be 201.6 kJ/kmol.K, equivalent to a specific heat capacity of 0.92 kJ/kg.K.

:<math>C_{m,p}(Na_2C_5O_7) = 2 \times C_{m,p}(Na) + 5 \times C_{m,p}(C) + 3.5 \times C_{m,p}(O_2) </math>
:<math>c_p(Na_2C_5O_7) = \frac{C_{m,p}(Na_2C_5O_7)}{M} </math> &nbsp; &nbsp; where M is the molecular weight

By the same rule, the heat capacity of solid mixed oxides may also be estimated by adding the heat capacities of their component oxides.<sup>[2]</sup>

== SysCAD Species Example ==

As an exercise, we have taken 500 '''solid''' species from the [[Importing_Species_into_User_Species_Database#Importing_Species_from_other_SysCAD_Database|Default.93.db3 SysCAD database]] and calculated Cp(25) using both the relevant [[Species_Table_-_Specific_Heat_(Cp)#Cp_Equation_Format|SysCAD Cp equation]] and Kopp's law, then sorted the results by relative difference. The full data table can be found below, but the smallest and largest errors are summarised here:

<syntaxhighlight>
                            Cmp(kJ/kmol.K)
 Species                    Kopp's Default  Diff(%) Cp(kJ/kg.K) 
 --------------------------------------------------------------
 Cu2Mg                       73.8    73.8    0.013   0.49
 CaO.MgO                     79.5    79.6    0.044   0.83
 MnSO4.5H2O                 325.8   325.7    0.052   1.35
 Cu6PS5I                    310.7   310.5    0.056   0.44
 NiSi2                       66.1    66.2    0.137   0.58
 CoO.Al2O3                  132.0   132.2    0.174   0.75
 .......
 Fe2S3                      117.9    47.8   59.468   0.57
 Na2C2O4                    132.0    34.0   74.248   0.99
 2CaO.[Al2O3].8SiO2         567.1    97.7   82.781   0.82
 CrO2                        52.7    99.7   89.232   0.63
 Ca3Al2Si3O12               360.5  1369.4  279.898   0.80
 Ca2Al2SiO7                 221.8   851.4  283.839   0.81
 MgCO3.3H2O                 208.4 -3689.2 1870.492   1.51
</syntaxhighlight>

{{collapse | title = Full Data Set | content = <syntaxhighlight>
                            Cmp(kJ/kmol.K)
 Species                    Kopp's Default  Diff(%) Cp(kJ/kg.K) 
 --------------------------------------------------------------
 Cu2Mg                       73.8    73.8    0.013   0.49
 CaO.MgO                     79.5    79.6    0.044   0.83
 MnSO4.5H2O                 325.8   325.7    0.052   1.35
 Cu6PS5I                    310.7   310.5    0.056   0.44
 NiSi2                       66.1    66.2    0.137   0.58
 CoO.Al2O3                  132.0   132.2    0.174   0.75
 CoS                         47.4    47.5    0.189   0.52
 Mg4Si6O21H12               701.5   702.9    0.210   1.14
 CaMg3[CO3]4                310.1   309.4    0.231   0.88
 Na8Fe2O7                   378.0   379.0    0.256   0.93
 FeSi                        45.1    45.2    0.269   0.54
 CaSO3.2H2O                 179.2   178.7    0.306   1.15
 Na2S2O3                    145.5   146.0    0.347   0.92
 CaCO3.MgCO3                155.3   155.9    0.403   0.84
 MnSn2                       80.3    80.0    0.437   0.27
 FeO.SiO2                    89.1    89.5    0.438   0.68
 Ag2SO4                     132.1   131.5    0.453   0.42
 SnSO4                      108.3   108.8    0.486   0.50
 CaSO3                       91.9    91.5    0.498   0.77
 2CaO*SiO2                  129.3   128.6    0.547   0.75
 KFe3[AlSi3O10][OH]2        393.7   391.6    0.550   0.77
 Mn3Si                       99.0    99.6    0.608   0.51
 Ag2O                        65.5    65.9    0.628   0.28
 SnS                         49.6    49.3    0.663   0.33
 Mg5[OH]2[CO3]4.4H2O        567.5   571.3    0.666   1.21
 Fe2B                        61.2    60.8    0.669   0.50
 Ca2B2O5                    146.1   147.1    0.687   0.80
 AgP2                        73.1    72.6    0.707   0.43
 Na2S2O3.5H2O               363.7   361.0    0.749   1.47
 CoSi2                       64.8    65.3    0.751   0.56
 CaMg2                       75.1    75.7    0.764   0.85
 MnSi                        46.3    45.9    0.844   0.56
 Al4Ca                      122.4   121.3    0.851   0.83
 ZnO                         40.1    40.4    0.870   0.49
 NiS2                        71.3    70.6    0.919   0.58
 Ca3B2O6                    186.1   187.9    0.970   0.78
 Na2S3                      124.1   125.3    1.004   0.87
 Fe2SiO4                    128.8   130.2    1.053   0.63
 Sn3S4                      171.4   169.5    1.056   0.35
 CoSn                        51.8    51.2    1.090   0.29
 CoSi                        44.8    44.3    1.098   0.51
 Mn2SiO4                    131.3   129.9    1.104   0.65
 CaS                         47.9    47.4    1.114   0.66
 Ag3PO4                     158.7   156.9    1.124   0.38
 MgO2                        54.2    53.6    1.138   0.96
 2CaO.2Al2O3.8SiO2.7H2O     965.2   954.0    1.163   1.05
 NaOH*                       57.3    58.0    1.205   1.43
 NaOH                        57.3    58.0    1.205   1.43
 CuS                         47.0    47.6    1.290   0.49
 NiO.Al2O3                  133.3   131.6    1.295   0.75
 AuI                         52.5    51.8    1.298   0.16
 3CaO.Al2O3                 212.5   209.7    1.308   0.79
 Ni7Si13                    442.5   436.7    1.310   0.57
 3CaO.SiO2                  169.3   171.6    1.349   0.74
 Sn2S3                      121.8   120.1    1.351   0.37
 FeSi2                       65.1    64.2    1.362   0.58
 Fe2P                        74.0    75.0    1.421   0.52
 CuP2                        72.1    71.1    1.431   0.57
 CuFeO2                      78.8    80.0    1.462   0.52
 Cu2O                        63.5    62.6    1.490   0.44
 Mg2[OH]2CO3.3H2O           291.6   296.1    1.533   1.48
 Mn5Si3                     191.6   194.6    1.536   0.53
 MgB4                        69.1    70.2    1.539   1.02
 Ni2Si                       72.2    71.0    1.597   0.50
 K2O                         73.2    72.0    1.668   0.78
 Al2Ca                       73.8    72.6    1.684   0.79
 CaZn2                       76.1    74.8    1.702   0.45
 3Na2O.2SiO2                311.6   306.2    1.717   1.02
 MgCl2.6H2O                 320.7   315.0    1.758   1.58
 AlClO                       55.9    56.9    1.829   0.71
 AuSn                        52.3    51.3    1.860   0.17
 AuCu3                       98.6    96.8    1.884   0.25
 Fe3P                        99.0   101.0    1.995   0.50
 CaB2O4                     106.1   104.0    1.996   0.84
 AuCu                        49.7    48.7    2.002   0.19
 Na2CO3                     108.8   111.0    2.017   1.03
 MnS2                        71.5    70.1    2.024   0.60
 Au2P3                      122.1   119.6    2.050   0.25
 CuFeS2                      94.7    96.6    2.053   0.52
 MgB2                        47.0    48.0    2.060   1.02
 Mn3O4                      137.7   140.5    2.076   0.60
 MnS                         48.9    49.9    2.080   0.56
 Mn2O3                       96.7    98.7    2.132   0.61
 CaAl2Si4O12.4H2O           504.4   493.6    2.141   1.07
 ZnSO4.H2O                  150.3   153.6    2.155   0.84
 FeSiO3                      89.1    91.0    2.214   0.68
 Ca3Si2O7                   218.6   213.7    2.233   0.76
 3CaO.2SiO2                 218.6   213.7    2.259   0.76
 Co2Al9                     268.0   261.9    2.274   0.74
 Ni3S4                      168.7   164.8    2.278   0.55
 3CaO.MgO.2SiO2             258.2   252.3    2.289   0.79
 Na2SO3                     122.9   120.1    2.291   0.98
 ZnFe2O4                    134.2   137.3    2.327   0.56
 FeNaO2                      82.6    84.5    2.379   0.74
 Fe3Si                       95.2    97.5    2.430   0.49
 MgCO3                       77.4    75.5    2.444   0.92
 ZnSO4.2H2O                 194.0   198.7    2.461   0.98
 CaSi                        45.3    46.4    2.463   0.66
 FeO.Cr2O3                  130.4   133.7    2.505   0.58
 CoS2                        70.0    68.2    2.534   0.57
 MnO2                        55.7    54.2    2.567   0.64
 FeCr2O4                    130.4   133.8    2.604   0.58
 ZnO.2ZnSO4                 253.4   246.6    2.685   0.63
 Fe3Al2Si3O12               359.7   350.0    2.701   0.72
 CoSO4                      106.1   103.2    2.722   0.68
 Co2B                        60.6    59.0    2.747   0.47
 ZnCO3                       77.9    80.1    2.748   0.62
 Na2CrO4                    138.3   142.1    2.766   0.85
 Mg2Si                       69.8    67.8    2.800   0.91
 Na2S2                      101.5    98.6    2.827   0.92
 NaS                         50.7    49.3    2.828   0.92
 2CaO.MgO.2SiO2             218.2   212.0    2.830   0.80
 K2CO3                      111.1   114.2    2.834   0.80
 CaZn                        50.7    49.3    2.835   0.48
 SnS2                        72.2    70.1    2.840   0.39
 Ca3Fe2Si3O12               362.1   351.8    2.846   0.71
 NiSi                        46.1    44.8    2.852   0.53
 ZnSO4.6H2O                 368.5   358.0    2.871   1.37
 SnI4                       135.9   132.0    2.876   0.22
 CdS                         48.8    47.3    2.878   0.34
 KMnO4                      114.3   117.6    2.907   0.72
 Al2S3                      116.3   112.9    2.920   0.77
 Na2O                        71.0    68.9    2.923   1.14
 2NiO.SiO2                  130.8   127.0    2.926   0.62
 Fe7Si8O22[OH]2             716.4   695.3    2.945   0.72
 Cu2[OH]2CO3                159.7   164.5    2.961   0.72
 CrO3                        67.3    69.3    2.966   0.67
 Zn3[PO4]2                  241.2   234.1    2.976   0.62
 Ca2Si                       70.6    72.7    2.990   0.65
 CoAl3                       97.6    94.5    3.175   0.70
 CuO.CuSO4                  144.8   140.2    3.182   0.61
 2FeO.SiO2                  128.8   132.9    3.182   0.63
 NiS                         48.7    47.1    3.229   0.54
 CaI2                        79.8    77.2    3.252   0.27
 CuI                         51.7    53.4    3.292   0.27
 Cu5FeS4                    237.7   245.6    3.346   0.47
 MgSO4.H2O                  149.8   144.8    3.356   1.08
 MgCl2.4H2O                 233.4   241.2    3.374   1.40
 MnCO3                       78.9    81.5    3.388   0.69
 CoB                         35.9    34.6    3.396   0.51
 2Na2O.SiO2                 191.3   184.7    3.420   1.04
 AuSn4                      133.2   137.9    3.467   0.20
 K2O.SiO2                   122.6   118.3    3.507   0.79
 CoCO3                       77.3    80.1    3.536   0.65
 NiP2                        73.8    71.1    3.626   0.61
 Al5Co2                     170.9   164.7    3.633   0.68
 SnI2                        81.4    78.4    3.659   0.22
 NaPO3                       96.0    92.5    3.673   0.94
 Na2S4                      146.7   152.2    3.729   0.84
 NaS2                        73.3    76.1    3.729   0.84
 MgB12                      157.6   151.6    3.782   1.02
 Mn2P                        76.5    73.6    3.840   0.54
 NaBO2                       68.5    65.9    3.843   1.04
 NiI2                        80.5    77.4    3.884   0.26
 Ag2S                        73.4    76.2    3.892   0.30
 Na2O.B2O3                  137.1   131.7    3.924   1.04
 CaSO4*2H2O                 193.9   186.2    3.965   1.13
 Ni3B                        89.3    85.7    3.999   0.48
 CoSO4.[6H2O]               367.9   353.0    4.060   1.40
 Cu3[OH]2[CO3]2             236.7   246.6    4.175   0.69
 Na4SiO4                    191.3   183.2    4.205   1.04
 Na2O2                       85.6    89.2    4.206   1.10
 FeSO4.7H2O                 411.9   394.5    4.221   1.48
 MgS                         47.5    45.5    4.222   0.84
 AlNi3                      102.5    98.1    4.325   0.50
 NiAl3                       98.9    94.6    4.357   0.71
 MnSiO3                      90.3    86.4    4.359   0.69
 Na2S                        78.9    82.3    4.366   1.01
 CaSi2                       65.3    68.2    4.376   0.68
 CaO.MgO.SiO2               128.9   123.2    4.390   0.82
 2CoO.SiO2                  128.3   133.9    4.439   0.61
 CoI2                        79.2    75.7    4.445   0.25
 Ca5[PO4]3OH                403.2   385.1    4.502   0.80
 Na2CO3.H2O                 152.5   145.6    4.510   1.23
 CaMgSiO4                   128.9   123.1    4.512   0.82
 CaSiO3                      89.3    85.3    4.528   0.77
 CaAl2Si4O12.2H2O           417.1   398.2    4.532   0.96
 3Ca3[PO4]2.Ca[OH]2         806.5   769.9    4.543   0.80
 Ni3S2                      123.5   117.7    4.638   0.51
 Ag2O2                       80.1    83.8    4.641   0.32
 Ca[OH]2                     83.6    87.5    4.658   1.13
 ZnS                         48.0    45.8    4.667   0.49
 AlB12                      156.9   149.6    4.689   1.00
 Zn2SiO4_gamma              129.5   123.3    4.733   0.58
 MgNi2                       77.1    73.4    4.755   0.54
 Ni6P5                      275.7   262.2    4.902   0.54
 NiP3                        97.6   102.5    4.971   0.64
 Sn[SO4]2                   189.5   180.0    5.027   0.61
 CaO                         40.0    42.0    5.044   0.71
 CoP3                        96.3   101.2    5.087   0.63
 Ni2B                        63.2    60.0    5.113   0.49
 ZnSiO3                      89.4    84.8    5.189   0.63
 Cr8O21                     494.7   520.6    5.235   0.66
 CoSO4.[7H2O]               411.6   390.0    5.244   1.46
 FeI2                        79.5    83.7    5.246   0.26
 CaFe[SiO3]2                178.4   168.9    5.329   0.72
 Ni3Sn                      105.3    99.6    5.372   0.36
 AlCo                        49.1    46.4    5.377   0.57
 6CaO.6SiO2.H2O             579.5   548.3    5.383   0.81
 MgSO4.6H2O                 368.0   348.2    5.391   1.61
 Ca3[PO4]2                  241.0   227.8    5.461   0.78
 FeSO4                      106.3   100.4    5.576   0.70
 SiS                         42.6    45.0    5.601   0.71
 Zn3P2                      123.9   116.9    5.662   0.48
 MgI2                        79.3    74.8    5.663   0.29
 NaI                         55.4    52.2    5.668   0.37
 CaC2O4.H2O                 144.7   153.0    5.747   0.99
 FeCO3                       77.6    82.1    5.776   0.67
 CuMg2                       74.2    69.9    5.789   0.66
 FeS                         47.7    50.4    5.812   0.54
 AuSn2                       79.3    83.9    5.838   0.18
 FeSi_alpha                  45.1    47.7    5.856   0.54
 AlB2                        46.4    43.6    5.894   0.95
 Zn2SiO4                    129.5   121.8    5.902   0.58
 Ca3P2                      123.6   116.3    5.910   0.68
 MgO                         39.6    37.2    5.949   0.98
 CaHPO4.2H2O                209.6   196.9    6.039   1.22
 3CaO.[2SiO2].3H2O          349.5   328.4    6.053   1.02
 Cu[OH]2                     82.8    87.8    6.159   0.85
 KHCO3                       96.3    90.4    6.165   0.96
 NaFe[SiO3]2                181.2   169.9    6.240   0.78
 Mn3P                       102.8   109.3    6.267   0.53
 Cu6PS5Cl                   300.4   319.4    6.336   0.49
 K2O.2SiO2                  171.9   160.9    6.361   0.80
 CaSO4                      106.6    99.6    6.506   0.78
 AgIO3                       96.6   102.9    6.536   0.34
 KI                          56.5    52.8    6.546   0.34
 CuSO4                      105.7    98.8    6.565   0.66
 Ca2P2O7                    201.0   187.8    6.576   0.79
 FeAl2O4                    132.3   123.5    6.598   0.76
 CuI2                        78.9    84.1    6.616   0.25
 MnO.Al2O3                  133.5   124.6    6.670   0.77
 Mg[OH]2                     83.2    77.6    6.687   1.43
 MnI2                        80.8    75.3    6.719   0.26
 Cr5O12                     292.7   312.5    6.750   0.65
 NiB                         37.1    34.6    6.762   0.53
 Cu2O.Fe2O3                 157.7   168.4    6.771   0.52
 Cu2S                        71.5    76.3    6.779   0.45
 Ni4B3                      137.5   128.1    6.835   0.51
 Na2SO4                     137.6   128.1    6.852   0.97
 Mn3C                        87.5    93.5    6.863   0.49
 CoP                         48.6    45.3    6.881   0.54
 MnSO4                      107.6   100.2    6.908   0.71
 Fe5Si3                     185.3   198.2    6.940   0.51
 Ni2Al3                     125.0   116.3    6.946   0.63
 Ag2CrO4                    132.8   142.1    7.047   0.40
 Na2O.SiO2                  120.3   111.8    7.069   0.99
 Ca2Al2Si3O10[OH]2          364.1   338.2    7.126   0.88
 ZnSO4                      106.7    99.1    7.136   0.66
 KClO4                      104.9   112.4    7.155   0.76
 CaCO3                       77.8    83.5    7.230   0.78
 AgP3                        96.9   103.9    7.233   0.48
 NaClO4                     103.8   111.3    7.281   0.85
 Co3O4                      133.1   123.3    7.320   0.55
 FePO4.2H2O                 194.9   180.5    7.353   1.04
 2CaO.Al2O3.SiO2            221.8   205.4    7.392   0.81
 CaAl2Si2O7[OH]2.H2O        318.5   294.9    7.412   1.01
 NaIO3                       99.4    92.0    7.418   0.50
 Fe2MgO4                    133.7   143.8    7.514   0.67
 MgSO4.7H2O                 411.7   380.7    7.516   1.67
 Na2O.2SiO2                 169.6   156.8    7.543   0.93
 Cu2SO4                     130.1   120.3    7.578   0.58
 SnO2                        56.3    52.0    7.682   0.37
 Ca2Sn                       77.6    71.6    7.752   0.39
 FeB                         36.1    38.9    7.753   0.54
 MnSO4.7H2O                 413.1   380.7    7.835   1.49
 Co2Si                       69.6    75.0    7.849   0.48
 FeO[OH]                     68.9    74.3    7.879   0.78
 Fe2NiO4                    134.9   145.6    7.935   0.58
 NaHCO3                      95.2    87.6    7.944   1.13
 CaSn                        52.3    48.1    7.975   0.33
 Na2B4O7                    203.2   186.9    8.015   1.01
 ZnSO4.7H2O                 412.2   379.2    8.016   1.43
 Ca2FeAl2Si3O12OH           389.4   358.1    8.021   0.81
 Ag2CO3                     103.3   111.6    8.036   0.37
 Na6P2O8                    333.9   306.7    8.159   1.02
 CuO                         39.1    42.3    8.171   0.49
 AgI                         52.6    56.9    8.186   0.22
 Na4P2O7                    262.9   241.3    8.243   0.99
 CaB4O7                     172.2   157.9    8.288   0.88
 CuCO3                       77.0    83.4    8.365   0.62
 Ca2Al2SiO6[OH]2            265.5   243.2    8.389   0.91
 Mg2SiO4                    128.5   117.7    8.398   0.91
 2CaO.5MgO.8SiO2.H2O        716.1   655.6    8.443   0.88
 NaAlSi2O6.H2O              224.1   204.8    8.589   1.02
 CaO*Al2O3*2SiO2            231.2   211.3    8.591   0.83
 NaAlSi3O8_gamma            229.7   209.9    8.637   0.88
 NiO                         40.8    44.3    8.650   0.55
 CaAl2SiO6                  181.8   166.0    8.731   0.83
 NiAl                        50.4    46.0    8.734   0.59
 Mg3Si2O5[OH]4              304.7   277.9    8.781   1.10
 3CaO.Al2O3.6H2O            474.3   432.6    8.804   1.25
 MnSO4.H2O                  151.2   137.9    8.819   0.89
 CaAl2Si2O8                 231.2   210.7    8.867   0.83
 CaO.Al2O3.SiO2             181.8   165.7    8.879   0.83
 AlO                         38.9    35.5    8.936   0.91
 MgCl2.2H2O                 146.1   159.2    8.986   1.11
 Na2B8O13                   335.4   304.9    9.107   0.99
 Na2P2O6                    192.0   174.5    9.125   0.94
 MgSO4.2H2O                 193.5   175.7    9.167   1.24
 MnO                         41.0    44.8    9.186   0.58
 SiP                         43.8    39.8    9.250   0.74
 KAlSi3O8_glass             230.9   209.4    9.309   0.83
 K2O.Al2O3.2SiO2            264.4   239.8    9.310   0.84
 Mg3Al2Si3O12               359.2   325.5    9.395   0.89
 KAlSiO4                    132.2   119.8    9.423   0.84
 KAl[SO4]2.3H2O             347.0   314.3    9.423   1.11
 K2O.Al2O3.4SiO2            363.1   328.6    9.495   0.83
 Ca2Mg5Si8O24H2             716.1   648.0    9.502   0.88
 Na2B6O10                   269.3   243.7    9.524   0.99
 NaB3O5                     134.7   121.8    9.524   0.99
 KMg3[AlSi3O10][OH]2        393.2   355.7    9.552   0.94
 MnSO4.4H2O                 282.2   255.2    9.552   1.27
 CaO.Al2O3                  132.5   119.8    9.574   0.84
 KAlSi2O6                   181.5   164.1    9.583   0.83
 Cu3P                        97.2    87.8    9.628   0.44
 NiCO3                       78.6    86.2    9.630   0.66
 ZnO.Cr2O3                  130.7   143.4    9.651   0.56
 NiSO4.H2O                  151.0   136.4    9.680   0.87
 SiO2                        49.3    44.5    9.707   0.82
 MgSO4                      106.2    95.8    9.759   0.88
 Fe[OH]3                    112.5   101.5    9.817   1.05
 Mg2Al3[AlSi5O18]H2O        554.5   500.0    9.831   0.92
 ZnO.Al2O3                  132.6   119.4    9.984   0.72
 NaCa2Mg4Al3Si6O24H2        752.1   676.7   10.029   0.90
 NaAlO2                      81.8    73.5   10.051   1.00
 Na2O.Al2O3                 163.5   147.1   10.061   1.00
 3MgO.[2SiO2].2H2O          304.7   273.9   10.093   1.10
 Mg48Si34O85[OH]62         4929.5  4431.4   10.104   1.09
 Na3PO4                     167.0   150.0   10.133   1.02
 NiSO4.4H2O                 281.9   253.1   10.217   1.24
 Mn2B                        63.7    57.2   10.227   0.53
 CuSO4.H2O                  149.4   134.0   10.285   0.84
 3CaO.[Al2O3].3SiO2         360.5   323.1   10.354   0.80
 MnO.Fe2O3                  135.1   149.2   10.416   0.59
 MgSO4.4H2O                 280.8   251.2   10.534   1.46
 CaHPO4                     122.3   109.4   10.544   0.90
 Na2O.Cr2O3                 161.6   178.8   10.616   0.76
 Fe2MnO4                    135.1   149.5   10.660   0.59
 KAl[SO4]2                  216.1   192.9   10.711   0.84
 Fe2[SO4]3                  293.9   262.4   10.737   0.74
 NaAlSi3O8                  229.7   204.8   10.835   0.88
 Na2O.Al2O3.6SiO2           459.5   409.7   10.836   0.88
 Al2[SO4]3.6H2O             554.2   494.1   10.841   1.23
 Na2SO4.7H2O                443.1   395.0   10.849   1.65
 NaAlSi3O8_alpha            229.7   204.8   10.850   0.88
 Na2Mg3Al2Si8O22[OH]2       720.5   641.6   10.947   0.92
 NaBH4                       97.2    86.4   11.026   2.57
 Mg3[PO4]2                  239.7   213.1   11.094   0.91
 K2O.4SiO2                  270.5   300.7   11.161   0.81
 KAl2[AlSi3O10][OH]2        367.0   325.9   11.198   0.92
 NiSO4.6H2O                 369.2   327.9   11.204   1.40
 2CaO.Fe2O3                 174.1   193.6   11.223   0.64
 Al2[SO4]3                  292.3   259.4   11.270   0.85
 Fe2O3                       94.1   104.8   11.299   0.59
 Ca[H2PO4]2.H2O             291.9   258.8   11.342   1.16
 CuO.Fe2O3                  133.2   148.4   11.356   0.56
 Al4B2O9                    251.2   222.6   11.364   0.92
 Na2O.Al2O3.4SiO2           360.8   319.8   11.380   0.89
 NaAlSi2O6                  180.4   159.9   11.380   0.89
 KClO3                       90.2   100.5   11.399   0.74
 Al4Mg2Si5O18               510.9   452.3   11.460   0.87
 Zn[OH]2                     83.7    74.1   11.488   0.84
 FeS2                        70.3    62.2   11.516   0.59
 Na2O.Al2O3.2SiO2           262.2   231.6   11.649   0.92
 NaAlSiO4                   131.1   115.8   11.649   0.92
 NiSO4.7H2O                 412.9   364.6   11.693   1.47
 Na2HPO4                    153.3   135.3   11.733   1.08
 Co2P                        73.4    64.8   11.736   0.49
 KCl                         46.2    51.7   11.877   0.62
 Fe7S8                      356.2   398.6   11.880   0.55
 CaO.2Al2O3                 225.1   198.2   11.943   0.87
 NaCl                        45.1    50.5   11.947   0.77
 Al2Si4O10[OH]2             333.5   293.7   11.948   0.93
 CaAl4Si2O10[OH]2           367.4   323.1   12.050   0.92
 7MgO.8SiO2.H2O             715.3   629.0   12.061   0.92
 AlCl3.6H2O                 337.0   296.2   12.092   1.40
 KAlSi3O8                   230.9   202.9   12.110   0.83
 CaMgSi2O6                  178.2   156.6   12.118   0.82
 NiFe2Cl8                   211.8   237.5   12.129   0.47
 CaO.Cr2O3                  130.7   146.7   12.242   0.63
 K2O.Fe2O3                  167.4   187.9   12.247   0.66
 MgO.Al2O3                  132.1   115.9   12.250   0.93
 Fe3O4                      133.9   150.3   12.294   0.58
 AgO                         40.1    45.0   12.383   0.32
 CaO.MgO.2SiO2              178.2   156.1   12.386   0.82
 NaClO3                      89.1   100.2   12.451   0.84
 MgCl2.H2O                  102.4   115.3   12.522   0.90
 NaAlCO3[OH]2               163.3   142.6   12.661   1.13
 Ni3Sn4                     186.2   162.5   12.714   0.29
 AlPO4                      106.8    93.0   12.899   0.88
 NaAl2[AlSi3O10][OH]2       365.9   318.5   12.967   0.96
 CuSO4.5H2O                 323.9   281.2   13.196   1.30
 NiO.Cr2O3                  131.4   148.8   13.216   0.58
 KAl3Si3O10[OH]2            367.0   318.3   13.292   0.92
 CuSO4.3H2O                 236.6   205.0   13.372   1.11
 KH                          43.8    37.9   13.403   1.09
 3Al2O3.2SiO2               376.3   325.4   13.509   0.88
 Ni3P                       102.1    87.8   14.007   0.49
 Mg3Si4O10[OH]2             359.7   308.9   14.121   0.95
 Al2O3.SiO2                 141.9   121.6   14.263   0.88
 CoO.Fe2O3                  133.6   152.8   14.399   0.57
 MgSiO3                      88.9   101.7   14.407   0.89
 SnO                         41.7    47.7   14.549   0.31
 CaO.Fe2O3                  134.1   153.6   14.562   0.62
 NaH                         42.6    36.4   14.630   1.78
 Al18B4O33                  965.1   823.2   14.703   0.91
 Ni2P                        76.0    64.8   14.742   0.51
 KAl3[OH]6[SO4]2            439.6   372.6   15.227   1.06
 P2O5                       121.0   102.4   15.398   0.85
 AuCl                        42.2    48.7   15.405   0.18
 Mg5Al2Si3O10[OH]8          612.9   517.3   15.611   1.10
 Na2H2P2O7                  235.6   198.2   15.904   1.06
 Na3AlCl6                   210.4   244.4   16.150   0.68
 SiI4                       128.9   108.0   16.167   0.24
 NaH2PO4                    139.6   116.9   16.312   1.16
 Fe[OH]2                     83.4    97.0   16.377   0.93
 AgClO3                      86.3   100.5   16.413   0.45
 K3AlCl6                    213.8   249.0   16.481   0.60
 Al2O3                       92.5    77.2   16.534   0.91
 AlP                         48.1    40.1   16.714   0.83
 Co[OH]2                     83.1    97.1   16.792   0.89
 K3Al2Cl9                   288.9   337.4   16.795   0.59
 [Al2O3].2SiO2              191.2   224.1   17.206   0.86
 Mg3P2                      122.4   101.0   17.435   0.91
 K2O.Al2O3.6SiO2            461.7   380.9   17.502   0.83
 KAlSi3O8_adular            230.9   190.5   17.502   0.83
 Ni3Sn2                     132.2   108.7   17.826   0.32
 Al2O3.3H2O                 223.5   183.5   17.899   1.43
 NiO.Fe2O3                  134.9   159.2   17.993   0.58
 SiS2                        65.2    77.5   18.866   0.71
 NiCl2                       60.0    71.7   19.507   0.46
 Au2O3                       94.6   114.0   20.540   0.21
 ZnCl2                       59.3    71.5   20.561   0.43
 CoO.Cr2O3                  130.1   157.2   20.811   0.57
 Ca2MgSi2O7                 218.2   172.7   20.851   0.80
 MgCl2                       58.8    71.2   21.024   0.62
 MnCl2                       60.2    73.0   21.177   0.48
 MgCl                        41.8    50.7   21.231   0.70
 AlCl3                       75.1    91.2   21.366   0.56
 Fe2Al4Si5O18               511.2   401.3   21.492   0.79
 AgClO2                      71.7    87.3   21.830   0.41
 Al2O3.H2O                  136.2   106.3   21.973   1.14
 Ca3MgSi2O8                 258.2   199.2   22.821   0.79
 Ni3C                        86.8   106.7   22.882   0.46
 CaCl2                       59.2    72.8   23.018   0.53
 CuCl2                       58.3    71.9   23.196   0.43
 CrCl3                       74.2    91.8   23.762   0.47
 Na2C2                       73.4    91.0   24.016   1.05
 CrCl2                       57.2    71.2   24.364   0.47
 AgCl                        42.3    52.7   24.432   0.30
 CaH2                        54.3    41.0   24.479   1.29
 AuCl3                       76.1    94.8   24.552   0.25
 ICl                         44.2    55.3   25.179   0.27
 NaO2                        57.5    72.1   25.481   1.05
 FeO                         39.7    49.9   25.618   0.55
 CuCl                        41.4    52.0   25.665   0.42
 CaO.[Al2O3].[2SiO2].2H2O   318.5   233.0   26.821   1.01
 FeCl3                       75.9    96.6   27.326   0.47
 Fe2O3.H2O                  137.8   175.7   27.538   0.78
 Fe3C                        83.7   107.1   27.946   0.47
 SnCl2                       60.9    78.0   28.197   0.32
 CaO.2SiO2.[2H2O]           225.9   162.1   28.252   1.06
 NiSO4                      107.4   138.0   28.540   0.69
 KAlCl4                     121.3   156.5   28.978   0.58
 NaAlCl4                    120.2   155.0   28.988   0.63
 FeCl2.4H2O                 233.5   301.2   28.992   1.17
 FeCl2                       59.0    76.3   29.468   0.47
 CaI                         52.5    36.5   30.468   0.31
 AuCl2                       59.2    77.3   30.559   0.22
 PCl5                       108.6   142.5   31.259   0.52
 Cr2O3                       90.7   119.8   32.116   0.60
 H2SO4.[3H2O]               241.2   318.9   32.242   1.59
 NaC2H3O2                   118.0    79.9   32.280   1.44
 MnCl3                       77.2   103.2   33.761   0.48
 CoCl2                       58.7    78.5   33.763   0.45
 H2SO4.[4H2O]               284.8   382.2   34.188   1.67
 MgH2                        53.9    35.3   34.433   2.05
 Co3S4                      164.8   221.6   34.467   0.54
 Mn3B4                      123.2    79.8   35.245   0.59
 CaOCl2                      73.9   100.3   35.737   0.58
 FeOCl                       56.7    77.0   35.903   0.53
 Si2H6                      126.9    79.0   37.758   2.04
 CoO                         39.5    54.9   39.182   0.53
 AlH3                        67.7    40.3   40.479   2.26
 C2Cl6                      118.7   171.5   44.481   0.50
 Cr[CO]6                    162.5   240.2   47.781   0.74
 PCl4                        91.6   138.1   50.698   0.53
 CaO2                        54.6    82.8   51.593   0.76
 CuH3                        67.9    30.3   55.390   1.02
 Fe2S3                      117.9    47.8   59.468   0.57
 Na2C2O4                    132.0    34.0   74.248   0.99
 2CaO.[Al2O3].8SiO2         567.1    97.7   82.781   0.82
 CrO2                        52.7    99.7   89.232   0.63
 Ca3Al2Si3O12               360.5  1369.4  279.898   0.80
 Ca2Al2SiO7                 221.8   851.4  283.839   0.81
 MgCO3.3H2O                 208.4 -3689.2 1870.492   1.51
</syntaxhighlight>}}

The relative error between the Kopp's law estimate and the default database is less than 7% for more than half of the data set, and only greater than 25% for 10% of the set. In fact, the largest error here (MgCO3.3H2O) has a ''negative'' Cp when calculated from the current default data, and some of the other very large errors look like the default Cp may be defective. It can be seen that testing via Kopp's Law is a good sanity check for any solid species. This approach also provides a method for estimating Cp when data is otherwise unavailable.

Note that the default SysCAD database continuously updated and is based on a range of open literature sources. As with all SysCAD default values, these are intended as a starting point for users, and we provide no guarantees for the veracity of the data.

The last column of the table shows the ''specific'' heat capacity calculated from the Kopp's law estimate (molar Cp divided by molecular weight). Across the data set, the values range from 0.17 to 2.57 kJ/kg.K. When no Cp data is supplied for a species in the [[Species Database]], SysCAD uses a value of 1.0 kJ/kg.K which appears to be a reasonable estimate. (For more information see [[Species Table - Specific Heat (Cp)]].)

Given the 25 kJ/kmol.K observation of the Dulong-Petit Law, it is noted that, when applying this estimation method with Kopp's law, if the average atomic weight of elements in the compound is about 25 kg/kmol, then the Cp would be about 1.0 kJ/kg.K. As such, the deviation from 1.0 kJ/kg.K is related to the average atomic weight of the elements in the species. For species with heavier atoms such as AuSn, the calculated value is lower at 0.17, while species with lighter atoms such as NaBH4 is higher at 2.57. Both of these extreme examples however still give a good correlation with their respective values calculated from the default database.

Plotting the distribution of all Kopp's law calculated specific heat capacities from this data set shows a mean value of ~0.75:

[[File:dulongp02.png|link=]]

==Conclusion==

Leveraging the Kopp-Neumann Law, you can estimate the heat capacity of any solid species. This can be useful for identification of faulty data or to estimate a value when no other information is available. SysCAD uses 1.0 kJ/kg.K as a default for solid species, which, while a reasonable estimate, is dependent on the atomic weights of the involved elements. Therefore, the Kopp's law estimate may be a better option than the default in some cases.

It is important to keep in mind that Kopp's law provides an '''estimate''' only, so should only be used as a fall-back for minor species in a project. The heat capacities of major components should be based on verified thermodynamic data and lab measurements.

From {{Available139|33288||y}}, the Kopp's law specific or molar heat capacity is shown alongside the calculated value for solid species on the [[SDB - Thermo1|Thermo1 tab]] of the [[Species Properties ($SDB)]] access window (''Species > View Properties'').

[[File:KoppExample.png|620px|link=]]

==References==
# (14 April 2023) ''"Dulong–Petit law"'', Wikipedia, Wikimedia Foundation, [https://en.wikipedia.org/wiki/Dulong%E2%80%93Petit_law https://en.wikipedia.org/wiki/Dulong-Petit_law]
# J Leitner, et al. (2003) "Estimation of heat capacities of solid mixed oxides", Thermochimica Acta, 395 27-46

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