Cluster - Populations |
Cluster populations are one of the most important results of the QCE scheme. They are determined iteratively for each p,T step. Particularly, they are necessary to calculate the partition function of the system Q from the partition functions of the single clusters qj. Consequently they are required for the calculation of thermodynamic data. The cluster populations specifies in which way 1 mole of monomers are distributed over the distinct cluster types. The analysis of these cluster populations provides information about the structural characteristics of the sample. Preferred structural motifs cause high populations of those clusters representing the motifs. Populations output can either be given in percent of the total amount of calculated clusters or as monomer normalized populations in fractions/percent of 1 mole.
Isobars |
The temperature-dependent volume at constant pressure is called an isobar. The volume depends largely on the QCE-parameters amf and bxv. These parameters can be optimized using an experimental isobar as follows: Peacemaker calculates the isobars for all user-defined combinations of parameters. Subsequently, these are compared to the experimental isobar and, using the least-mean-square method, the best one is chosen. The appropriate combination of parameters is assumed to be optimal and can now be applied in order to calculate thermodynamic data.
Entropy S |
The entropy S is a measure for the disorder in a system. In statistical physics there is a relation to the phase space volume. For the calculation of the entropy S the logarithm of the partition function Q and its first derivative are used. The calculated values are absolute entropies S. In order to ensure comparability to experimental data these values are referenced. The entropy S at any (optional) reference temperature is set to 0 J/(mol*K), all further values are referenced accordingly. Apart from referenced entropies, absolute ones as well as the parts calculated seperately from the molecular degrees of freedom can be given as output.
Internal Energy U |
The internal energy U of a system corresponds to its total energy. It is an extensive property, i.e. it depends on the amount of substance in the system. The internal energies U calculated by Peacemaker are molar energies, they refer to 1 mol, their unit is kJ/mol. In order to provide comparability with experimental data the obtained absolute values are referenced. At a (custom) reference temperature the internal energy U is set to 0 kJ/mol, all further values are changed accordingly. The internal energy U is calculated from the first derivative of the logarithm of the partition function Q.
Enthalpy H |
The enthalpy H may be defined as the sum of internal energy U and the pressure-volume work pV. The change of enthalpy of a system corresponds to the transferred heat at constant pressure. The output of this quantity is given as as a molar energy in kJ/mol.
The first derivative of the logarithm of the partition function Q as well as the term pV is used for calculating the enthalpy H. The pressure p is provided as input by the user and the volume V is a result of the QCE-iteration.
The calculated data are absolute values. They are referenced by subtracting the reference value of the internal energy.
Free Helmholtz Energy A |
The Helmholtz energy A is the difference of the internal Energy U and the product TS. The free energy change equals the amount of heat available at constant volume and temperature. It is calculated from the logarithm of the partition function Q. In analogy to the internal energy, the obtained absolute values are referenced. The absolute values as well as the referenced ones can be given as output.
Free Gibbs Energy G |
The Gibbs free energy G is the difference of enthalpy H and the product TS. It may also be written as the sum of Helmholtz free energy A and the expansion work pV. Whereas Helmholtz free energy is applied for processes at constant volume and temperature (isochoric and isothermal) Gibbs free energy is the appropriate quantity for processes at constant pressure and temperature (isobaric and isothermal). Within the QCE iteration Gibbs free energy is of special importance. Out of the several solutions resulting from the population polynomial and the volume polynomial the physically most meaningful is chosen by means of the Gibbs free energy. This one is used for further calculations. The calculation of Gibbs free energy is carried out using the first derivative of the logarithm of the partition function Q and adding the pV term. The pressure is given as input by the user and the volume is the result of the iteration. The obtained absolute values are referenced, too. Both absolute and referenced values can be given as output.
Heat Capacity Cv |
The heat capacity C is the measure of heat energy required to increase the temperature of a certain amount of substance by 1 K. There is a difference between an isobaric change of state (at constant pressure) and an isochoric change of state (at constant volume). Peacemaker calculates the isochoric heat capacity Cv. The amount of substance may be given in units of mass (g) or in the number of molecules (mol). The values calculated by Peacemaker are molar heat capacities, the unit is J/(mol*K). The quantity Cv depends on the first and the second derivative of the logarithm of the partition function Q with respect to temperature. These derivatives can be calculated analytically and numerically, respectively. Apart from the total isochoric heat capacity Peacemaker prints the decomposition into rotational, translational, vibrational and electronic degress of freedom. Finally, there is the possibility to use the partition function of the Morse Oscillator instead of the harmonic approximation and improve the results that way.