# Heat Capacities and Enthalpies of Normal Alkanes in an Ideal Gas State

^{*}

## Abstract

**:**

## 1. Introduction

_{n}H

_{2n + 2}, the following inequality holds:

_{n}H

_{2n+2}, with different numbers of carbon atoms, $n$, in the molecule. Knowledge of this dependence allows accurate estimates of the heat capacities and enthalpies of vapors of large n-alkanes with scarce or lacking data, thus providing the accurate interpolation/extrapolation of available experimental data without applying various semiempirical methods. Furthermore, they can be readily used in CFD applications instead of high-order polynomials with specific sets of coefficients for each compound.

## 2. Heat Capacity

_{2}group to the molecule. For $n\ge 4$, only CH

_{2}groups can be the nearest neighbors of the added group since the edge CH

_{3}groups already have such neighbors. In this case, the C

_{n+1}H

_{2n + 4}molecule differs from the C

_{n}H

_{2n + 2}molecule structurally in that the number of CH

_{2}groups in it that are not adjacent to the edge groups is larger by one. Therefore, it can be assumed that, up to interaction with more distant neighbors, the addition of the CH

_{2}group at $n\ge 4$ leads to an increase in the heat capacity $f\left(T\right)$ on the isotherm, which does not depend on $n$:

_{2}groups that would not be adjacent with an edge CH

_{3}group. Nevertheless, at $T\ge 500$ K, the error is also at the level of 0.1% on average.

## 3. Enthalpy

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${C}_{P}^{0}$ | ideal gas heat capacity |

${C}_{P,calc}^{0}$ | calculated heat capacity |

${C}_{P,exp}^{0}$ | measured heat capacity |

${C}_{P,G4}^{0}$ | heat capacity calculated by G4 method |

$\Delta {C}_{P}^{0}$ | error in the estimation of ${C}_{P}^{0}$ |

${C}_{P,PM6}^{0}$ | heat capacity calculated by PM6 method |

$f$ | function in Equation (3) |

$H$ | real gas enthalpy |

${H}^{0}$ | ideal gas enthalpy |

${H}_{res}$ | residual enthalpy |

${H}_{n,calc}^{0}$ | calculated ideal gas enthalpy |

${H}_{n,exp}^{0}$ | measured ideal gas enthalpy |

$\Delta {H}_{n}^{0}$ | error in the estimation of ${H}^{0}$ |

$n$ | numbers of carbon atoms in n-alkane molecules |

$P$ | pressure |

$T$ | temperature |

${T}_{1}$ | reference temperature |

$V$ | volume |

$\tau $ | reduced temperature |

## Appendix A

$\mathit{T},\text{}\mathbf{K}$ | ${\mathit{C}}_{\mathit{P},5}^{0}\phantom{\rule{0ex}{0ex}}\mathbf{J}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ | ${\mathit{C}}_{\mathit{P},6}^{0}\phantom{\rule{0ex}{0ex}}\mathbf{J}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ |
---|---|---|

[2] | [2] | |

300 | 120.62 | 143.26 |

400 | 152.55 | 181.54 |

500 | 182.59 | 217.28 |

700 | 231.38 | 274.05 |

900 | 266.94 | 315.06 |

1100 | 293.72 | 345.18 |

1300 | 313.80 | 368.19 |

1500 | 330.54 | 389.11 |

**Table A2.**Accuracy of Equation (5) for ${C}_{P}^{0}\left(T\right)$ of n-C

_{7}H

_{16}and n-C

_{10}H

_{22}[2].

$\mathit{T},\text{}\mathbf{K}$ | $\mathit{f}\left(\mathit{T}\right)\phantom{\rule{0ex}{0ex}}\mathbf{J}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ | ${\mathit{C}}_{\mathit{P},7}^{0}\phantom{\rule{0ex}{0ex}}\mathbf{J}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ | ${\mathit{C}}_{\mathit{P},7}^{0}\phantom{\rule{0ex}{0ex}}\mathbf{J}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ | $\mathbf{\Delta}{\mathit{C}}_{\mathit{P},7}^{0},\text{}\%$ | ${\mathit{C}}_{\mathit{P},10}^{0}\phantom{\rule{0ex}{0ex}}\mathbf{J}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ | ${\mathit{C}}_{\mathit{P},10}^{0}\phantom{\rule{0ex}{0ex}}\mathbf{J}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ | $\mathbf{\Delta}{\mathit{C}}_{\mathit{P},10}^{0}$ |
---|---|---|---|---|---|---|---|

Equation (4) | Equation (5) | [2] | Equation (5) | [2] | |||

300 | 22.64 | 165.90 | 165.98 | −0.05 | 233.82 | 234.18 | −0.15 |

400 | 28.99 | 210.53 | 210.66 | −0.06 | 297.50 | 297.98 | −0.16 |

500 | 34.69 | 251.97 | 252.09 | −0.05 | 356.04 | 356.43 | −0.11 |

700 | 42.67 | 316.72 | 317.15 | −0.13 | 444.73 | 446.43 | −0.38 |

900 | 48.12 | 363.18 | 363.59 | −0.11 | 507.54 | 508.36 | −0.16 |

1100 | 51.46 | 396.64 | 397.06 | −0.10 | 551.02 | 551.87 | −0.15 |

1300 | 54.39 | 422.58 | 422.58 | 0 | 585.75 | 585.76 | 0 |

1500 | 58.57 | 447.68 | 443.50 | 0.9 | 623.39 | 610.86 | 2.05 |

## Appendix B

$\mathit{T}$ | ${\mathit{C}}_{\mathit{P},5}^{0}\phantom{\rule{0ex}{0ex}}\mathbf{cal}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ | ${\mathit{C}}_{\mathit{P},6}^{0}\phantom{\rule{0ex}{0ex}}\mathbf{cal}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ | $\mathit{f}\left(\mathit{T}\right)\phantom{\rule{0ex}{0ex}}\mathbf{cal}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ | |||
---|---|---|---|---|---|---|

G4 | PM6 | G4 | PM6 | G4 | PM6 | |

300 | 26.67 | 27.34 | 31.56 | 32.30 | 4.89 | 4.96 |

400 | 34.52 | 35.90 | 40.97 | 42.54 | 6.45 | 6.64 |

500 | 42.06 | 44.04 | 49.99 | 52.26 | 7.93 | 8.22 |

700 | 54.43 | 57.14 | 64.74 | 67.83 | 10.31 | 10.69 |

900 | 63.67 | 66.63 | 75.70 | 79.07 | 12.03 | 12.44 |

1100 | 70.63 | 73.53 | 83.93 | 87.22 | 13.30 | 13.69 |

1300 | 75.90 | 78.57 | 90.14 | 93.15 | 14.24 | 14.58 |

1500 | 79.91 | 82.28 | 94.84 | 97.52 | 14.93 | 15.24 |

**Table B2.**Accuracy of Equation (5) for ${C}_{P}^{0}\left(T\right)$ of n-C

_{4}H

_{10}, n-C

_{8}H

_{18}, n-C

_{10}H

_{22}[12].

$\mathit{T},\text{}\mathbf{K}$ | ${\mathit{C}}_{\mathit{P},4}^{0}\phantom{\rule{0ex}{0ex}}\mathbf{cal}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ | ${\mathit{C}}_{\mathit{P},4}^{0}\phantom{\rule{0ex}{0ex}}\mathbf{cal}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ | $\mathbf{\Delta}{\mathit{C}}_{\mathit{P},4}^{0}\text{}\%$ | ${\mathit{C}}_{\mathit{P},8}^{0}\phantom{\rule{0ex}{0ex}}\mathbf{cal}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ | ${\mathit{C}}_{\mathit{P},8}^{0}\phantom{\rule{0ex}{0ex}}\mathbf{cal}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ | $\mathbf{\Delta}{\mathit{C}}_{\mathit{P},10}^{0},\%$ | ${\mathit{C}}_{\mathit{P},10}^{0}\phantom{\rule{0ex}{0ex}}\mathbf{cal}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ | ${\mathit{C}}_{\mathit{P},10}^{0}\phantom{\rule{0ex}{0ex}}\mathbf{cal}/\left(\mathbf{mol}\text{\xb7}\mathbf{K}\right)$ | $\mathbf{\Delta}{\mathit{C}}_{\mathit{P},20}^{0}\%$ |
---|---|---|---|---|---|---|---|---|---|

Equation (5) | G4 | Equation (5) | G4 | Equation (5) | PM6 | ||||

300 | 21.78 | 21.76 | 0.09 | 41.34 | 41.34 | 0 | 52.14 | 52.17 | −0.06 |

400 | 28.07 | 28.05 | 0.07 | 53.87 | 53.89 | −0.04 | 69.10 | 69.13 | −0.04 |

500 | 34.13 | 34.12 | 0.03 | 65.85 | 65.87 | −0.03 | 85.14 | 85.13 | 0.01 |

700 | 44.12 | 44.14 | −0.05 | 85.36 | 85.33 | 0.04 | 110.59 | 110.60 | −0.01 |

900 | 51.64 | 51.67 | −0.06 | 99.76 | 99.70 | 0.06 | 128.83 | 128.84 | −0.01 |

1100 | 57.33 | 57.39 | −0.1 | 110.53 | 110.44 | 0.08 | 141.98 | 141.98 | 0 |

1300 | 61.66 | 61.72 | −0.1 | 118.62 | 118.52 | 0.08 | 151.47 | 151.51 | −0.02 |

1500 | 64.98 | 65.02 | −0.06 | 124.70 | 124.63 | 0.06 | 158.48 | 158.51 | −0.02 |

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$\mathit{T}$ | ${\mathit{C}}_{\mathit{P},5}^{0}$ cal/(mol·K) | ${\mathit{C}}_{\mathit{P},6}^{0}$ cal/(mol·K) | $\mathit{f}\left(\mathit{T}\right)$ cal/(mol·K) |
---|---|---|---|

[1] | [1] | Equation (4) | |

300 | 28.87 | 34.37 | 5.50 |

400 | 36.53 | 43.47 | 6.94 |

500 | 43.58 | 51.83 | 8.25 |

700 | 54.83 | 65.10 | 10.27 |

900 | 63.18 | 74.93 | 11.75 |

1100 | 69.48 | 82.32 | 12.84 |

1300 | 74.24 | 87.89 | 13.65 |

1500 | 77.83 | 92.10 | 14.27 |

**Table 2.**Accuracy of Equation (5) for ${C}_{P}^{0}\left(T\right)$ of n-C

_{4}H

_{10}, n-C

_{10}H

_{22}, n-C

_{20}H

_{42}.

$\mathit{T},\text{}\mathbf{K}$ | ${\mathit{C}}_{\mathit{P},4}^{0}$ cal/(mol·K) | ${\mathit{C}}_{\mathit{P},4}^{0}$ cal/(mol·K) | $\mathbf{\Delta}{\mathit{C}}_{\mathit{P},4}^{0}\text{}\%$ | ${\mathit{C}}_{\mathit{P},10}^{0}$ cal/(mol·K) | ${\mathit{C}}_{\mathit{P},10}^{0}$ cal/(mol·K) | $\mathbf{\Delta}{\mathit{C}}_{\mathit{P},10}^{0},$% | ${\mathit{C}}_{\mathit{P},20}^{0}$ cal/(mol·K) | ${\mathit{C}}_{\mathit{P},20}^{0}$ cal/(mol·K) | $\mathbf{\Delta}{\mathit{C}}_{\mathit{P},20}^{0}$% |
---|---|---|---|---|---|---|---|---|---|

Equation (5) | [1] | Equation (5) | [1] | Equation (5) | [1] | ||||

300 | 23.37 | 23.40 | −0.13 | 56.37 | 56.34 | 0.05 | 111.37 | 111.28 | 0.08 |

400 | 29.59 | 29.60 | −0.03 | 71.23 | 71.24 | 0.01 | 140.63 | 140.65 | −0.01 |

500 | 35.33 | 35.34 | −0.03 | 84.83 | 84.81 | 0.02 | 167.33 | 167.27 | 0.04 |

700 | 44.56 | 44.55 | 0.02 | 106.18 | 106.21 | −0.03 | 208.88 | 208.97 | −0.04 |

900 | 51.43 | 51.44 | −0.02 | 121.93 | 121.92 | 0 | 239.43 | 239.38 | 0.02 |

1100 | 56.64 | 56.64 | 0 | 133.68 | 133.7 | −0.02 | 262.08 | 262.0 | 0.03 |

1300 | 60.59 | 60.58 | 0.02 | 142.49 | 142.5 | −0.01 | 278.99 | 279.1 | −0.04 |

1500 | 63.56 | 63.57 | −0.02 | 149.18 | 149.2 | −0.01 | 291.88 | 291.9 | −0.01 |

$\mathit{T},\text{}\mathbf{K}$ | ${\mathit{C}}_{\mathit{P},3}^{0}$ cal/(mol·K) | $\mathbf{\Delta}{\mathit{C}}_{\mathit{P},3}^{0},\text{}\%$ | |
---|---|---|---|

Equation (5) | [1] | ||

300 | 17.87 | 17.66 | 1.2 |

400 | 22.65 | 22.54 | 0.5 |

500 | 27.08 | 27.04 | 0.1 |

700 | 34.29 | 34.20 | 0 |

900 | 39.68 | 39.61 | 0.3 |

1100 | 43.80 | 43.75 | 0.1 |

1300 | 46.94 | 46.89 | 0.1 |

1500 | 49.29 | 49.26 | 0.1 |

**Table 4.**Accuracy of Equation (7) for calculating ${H}_{n}^{0}\left({T}_{1}\right)$ with $n$ = 4–7, 10, 15, and 20 (${T}_{1}$ = 300 K).

$\mathit{n}$ | ${\mathit{H}}_{\mathit{n}}^{0}\left({\mathit{T}}_{1}\right),\text{}\mathbf{cal}/\mathbf{mol}$ | $\Delta {\mathit{H}}_{\mathit{n}}^{0}\left({\mathit{T}}_{1}\right),\text{}\%$ | |
---|---|---|---|

Equation (7) | [1] | ||

4 | 4679 | 4689 | −0.2 |

5 | 5682 | 5682 | 0 |

6 | 6685 | 6687 | 0 |

7 | 7688 | 7689 | 0 |

10 | 10,697 | 10,696 | 0 |

15 | 15,712 | 15,714 | 0 |

20 | 20,727 | 20,730 | 0 |

**Table 5.**Accuracy of Equations (8) and (9) for ${C}_{P}^{0}\left(T\right)$ of n-C

_{5}H

_{12}and n-C

_{6}H

_{14}.

$\mathit{T},\text{}\mathbf{K}$ | ${\mathit{C}}_{\mathit{P},5}^{0}$ cal/(mol·K) | ${\mathit{C}}_{\mathit{P},5}^{0}$ cal/(mol·K) | $\mathbf{\Delta}{\mathit{C}}_{\mathit{P},5}^{0},\text{}\%$ | ${\mathit{C}}_{\mathit{P},6}^{0}$ cal/(mol·K) | ${\mathit{C}}_{\mathit{P},6}^{0}$ cal/(mol·K) | $\mathbf{\Delta}{\mathit{C}}_{\mathit{P},6}^{0},\text{}\%$ |
---|---|---|---|---|---|---|

Equation (8) | [1] | Equation (9) | [1] | |||

300 | 28.87 | 28.87 | 0 | 34.34 | 34.37 | −0.1 |

400 | 36.65 | 36.53 | 0.3 | 43.61 | 43.47 | 0.3 |

500 | 43.53 | 43.58 | −0.1 | 51.76 | 51.83 | −0.1 |

700 | 54.79 | 54.83 | −0.1 | 65.06 | 65.10 | 0.1 |

900 | 63.24 | 63.18 | 0.1 | 75.01 | 74.93 | 0.1 |

1100 | 69.49 | 69.48 | 0 | 82.34 | 82.32 | 0 |

1300 | 74.17 | 74.24 | −0.1 | 87.82 | 87.89 | −0.1 |

1500 | 77.88 | 77.83 | 0.1 | 92.18 | 92.10 | 0.1 |

**Table 6.**Accuracy of Equation (13) for calculating ${H}_{n}^{0}\left(T\right)$ with $n$ = 4, 10, and 20.

$\mathit{T},\text{}\mathbf{K}$ | ${\mathit{H}}_{4}^{0},\mathbf{cal}/\mathbf{mol}$ | $\mathbf{\Delta}{\mathit{H}}_{4}^{0}\text{}\%$ | ${\mathit{H}}_{10}^{0},\mathbf{cal}/\mathbf{mol}$ | $\mathbf{\Delta}{\mathit{H}}_{10}^{0}$% | ${\mathit{H}}_{20}^{0},\mathbf{cal}/\mathbf{mol}$ | $\Delta {H}_{20}^{0}$% | |||
---|---|---|---|---|---|---|---|---|---|

Equation (13) | [1] | Equation (13) | [1] | Equation (13) | [1] | ||||

300 | 4679 | 4689 | −0.02 | 10,697 | 10,696 | 0 | 20,727 | 20,730 | 0.01 |

400 | 7337 | 7340 | 0 | 17,103 | 17,030 | 0.4 | 33,380 | 33,332 | 0.17 |

500 | 10,593 | 10,595 | 0 | 24,924 | 24,905 | 0.08 | 48,810 | 48,772 | 0.08 |

700 | 18,618 | 18,620 | 0 | 44,111 | 44,072 | 0.09 | 86,599 | 86,506 | 0.1 |

900 | 28,252 | 28,251 | 0 | 67,017 | 66,996 | 0.03 | 131,627 | 131,580 | 0.08 |

1100 | 39,091 | 39,083 | 0.02 | 92,663 | 92,620 | 0.05 | 181,949 | 181,850 | 0.05 |

1300 | 50,826 | 50,843 | −0.03 | 120,320 | 120,320 | 0 | 236,143 | 236,220 | 0.03 |

1500 | 63,248 | 63,270 | −0.03 | 149,522 | 149,500 | 0.01 | 293,311 | 293,240 | 0.02 |

**Table 7.**Accuracy of Equation (14 for ${C}_{P}^{0}\left(T\right)$ of n-C

_{4}H

_{10}, n-C

_{10}H

_{22}, and n-C

_{20}H

_{42}.

$\mathit{T},\text{}\mathbf{K}$ | ${\mathit{C}}_{\mathit{P},4}^{0}$ cal/(mol·K) | $\mathbf{\Delta}{\mathit{C}}_{\mathit{P},4}^{0}\text{}\%$ | ${\mathit{C}}_{\mathit{P},10}^{0}$ cal/(mol·K) | $\mathbf{\Delta}{\mathit{C}}_{\mathit{P},10}^{0}$% | ${\mathit{C}}_{\mathit{P},20}^{0}$ cal/(mol·K) | $\Delta {C}_{P20}^{0}$% | |||
---|---|---|---|---|---|---|---|---|---|

Equation (14) | [1] | Equation (14) | [1] | Equation (14) | [1] | ||||

300 | 23.34 | 23.40 | −0.26 | 56.35 | 56.34 | 0 | 111.38 | 111.28 | 0.09 |

400 | 29.69 | 29.60 | 0.3 | 71.45 | 71.24 | 0.3 | 141.04 | 140.65 | 0.3 |

500 | 35.30 | 35.34 | −0.1 | 84.67 | 84.81 | −0.16 | 166.95 | 167.27 | −0.2 |

700 | 44.52 | 44.55 | 0.07 | 106.15 | 106.21 | −0.1 | 208.88 | 208.97 | −0.04 |

900 | 51.47 | 51.44 | 0 | 122.09 | 121.92 | 0.1 | 239.78 | 239.38 | 0.17 |

1100 | 56.64 | 56.64 | 0 | 133.75 | 133.7 | 0 | 262.26 | 262.0 | 0.1 |

1300 | 60.52 | 60.58 | −0.1 | 142.42 | 142.5 | −0.1 | 278.94 | 279.1 | −0.06 |

1500 | 63.57 | 63.57 | 0 | 149.40 | 149.2 | 0.1 | 292.44 | 291.9 | 0.18 |

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**MDPI and ACS Style**

Kuznetsov, N.M.; Frolov, S.M.
Heat Capacities and Enthalpies of Normal Alkanes in an Ideal Gas State. *Energies* **2021**, *14*, 2641.
https://doi.org/10.3390/en14092641

**AMA Style**

Kuznetsov NM, Frolov SM.
Heat Capacities and Enthalpies of Normal Alkanes in an Ideal Gas State. *Energies*. 2021; 14(9):2641.
https://doi.org/10.3390/en14092641

**Chicago/Turabian Style**

Kuznetsov, Nikolai M., and Sergey M. Frolov.
2021. "Heat Capacities and Enthalpies of Normal Alkanes in an Ideal Gas State" *Energies* 14, no. 9: 2641.
https://doi.org/10.3390/en14092641