# Van der Waals equation

**Objective:**- To study the
*van der Waals*equation and to determine if it can explain the behaviour of a real gas. **Application:**- We'll use acetone as the system of study. Its critical point is known, and has the values:
**T**and_{c}= 508.10 K**P**._{c}= 46.35 atm

$\left(p+\frac{a}{{v}^{2}}\right)\left(v-b\right)=RT$

From the expressions for the critical coordinates provided by the model, we obtain:

${T}_{c}=\frac{8}{27}\frac{a}{Rb}$

${p}_{c}=\frac{1}{27}\frac{a}{{b}^{2}}$

and so, we can calculate the characteristic constants **a = 15.800111 atmL ^{2}/mol^{2}** and

**b = 0.112363**.

## P-V diagrams

From the real gas equation, we obtain:

$p=\frac{RT}{v-b}-\frac{a}{{v}^{2}}$

Using the previous equation, we can plot different isotherms in a p-v diagram that will allow us to visualize different situations of interest.

Taking an interval of molar volumes from 0 to 2 L/mol, we can use `pvdiagram.c`

(inside the folder `pvdiagrams/1_pvdiagram`

) to calculate the values of the associated pressure *p* for different temperatures. The program is prepared for the fact that measurements at different temperatures may have a different set of *v* data, but the results for all temperatures are dumped into a single output file (`output.txt`

).

Doing so, we obtain the following p-v diagram, showing the different isothermic curves for each temperature set:

v [L/mol] | p [atm] | ||||
---|---|---|---|---|---|

415 K | 460 K | 508,10 K | 530 K | ||

0.12 | 3360.534 | 3843.906 | 4360.470 | 4595.818 | |

0.13 | 995.339 | 1204.644 | 1428.322 | 1530.230 | |

0.14 | 425.697 | 559.268 | 702.011 | 767.046 | |

0.15 | 202.307 | 300.389 | 405.206 | 452.961 | |

0.16 | 97.462 | 174.954 | 257.768 | 295.498 | |

0.17 | 43.945 | 107.992 | 176.438 | 207.622 | |

0.18 | 15.675 | 70.254 | 128.580 | 155.153 | |

0.20 | -6.537 | 35.585 | 80.601 | 101.110 | |

0.22 | -10.164 | 24.132 | 60.783 | 77.481 | |

0.24 | -7.583 | 21.339 | 52.247 | 66.329 | |

0.26 | -3.137 | 21.867 | 48.588 | 60.762 | |

0.28 | 1.549 | 23.570 | 47.103 | 57.825 | |

0.30 | 5.878 | 25.552 | 46.577 | 56.156 | |

0.35 | 14.280 | 29.814 | 46.415 | 53.978 | |

0.40 | 19.607 | 32.441 | 46.156 | 52.404 | |

0.45 | 22.805 | 33.738 | 45.422 | 50.746 | |

0.50 | 24.624 | 34.147 | 44.324 | 48.961 | |

0.55 | 25.559 | 33.994 | 43.008 | 47.115 | |

0.60 | 25.925 | 33.495 | 41.585 | 45.271 | |

0.65 | 25.925 | 32.791 | 40.129 | 43.472 | |

0.75 | 25.302 | 31.091 | 37.278 | 40.097 | |

0.80 | 24.821 | 30.189 | 35.926 | 38.540 | |

0.90 | 23.717 | 28.403 | 33.412 | 35.694 | |

1.00 | 22.553 | 26.712 | 31.157 | 33.181 | |

1.50 | 17.511 | 20.172 | 23.015 | 24.310 | |

2.00 | 14.085 | 16.041 | 18.131 | 19.083 |

We are going to calculate the relevant thermodynamic information, i.e.:

- the p-v diagram and the Binodal Curve,
- the pv-p diagram and the Boyle temperature,
- and the Compressibility diagram.