Thermodynamic functions pdf

B The system in equilibrium state or moving from one equilibrium state to another equilibrium state. Which thermodynamic function accounts automatically for enthalpy and entropy both? B There is an exchange of energy as well as a matter between the system and the surroundings in a closed system. C The presence of reactants in a closed vessel made up of copper is an example of a closed system. D The presence of reactants in a thermos flask or any other closed insulated vessel is an example of a closed system.

Answer: The presence of reactants in a closed vessel made up of copper is an example of a closed system. In an adiabatic process, no transfer of heat takes place between system and surroundings. Choose the correct option for free expansion of an ideal gas under adiabatic condition from the following. The volume of gas is reduced to half from its original volume.

The bond energy in kcal mol -1 of a C-C single bond is approximately. In which of the following process, a maximum increase in entropy is observed? A system absorb 10 kJ of heat at constant volume and its temperature rises from C to C. The enthalpy of fusion of water is 1. The additional information required to determine the average energy for C — H bond formation would be. An ideal gas is allowed to expand both reversibly and irreversibly in an isolated system.

If Ti is the initial temperature and Tf is the final temperature, which of the following statements is correct? If one mole of ammonia and one mole of hydrogen chloride are mixed in a closed container to form ammonium chloride gas, then.

In a reversible process the system absorbs kJ heat and performs kJ work on the surroundings. What is the increase in the internal energy of the system? The entropy change for vaporisation is:. The expansion is carried out at K and K respectively. Choose the correct option. The spontaneity means, having the potential to proceed without the assistance of an external agency. The processes which occur spontaneously are. Also a new formula for the hadronic mass spectrum in terms of To and qo was derived from that theory and in Ref.

The non extensive self-consistent theory [17] imposes much more restrictive tests to the applicability of the Tsallis statistics in HEP and a number of analysis of experimental data [18—21] have shown that the theoretical predictions are in agreement with the experimental findings. In the present work we extend the thermodynamics to finite chemical potential systems, which is of importance in the study of nucleus-nucleus collisions and of astrophysical objects.

An important class of compact objects are protoneutron stars. The understanding of their evolution in time from the moment they are born as remnants of supernova explosions until they completely cool down to stable neutron stars, has been a matter of intense investigation. All sorts of phenomenological equations of state EOS , relativistic and non-relativistic ones, have been used to describe protoneutron star matter.

These EOS are normally parameter dependent and are adjusted so as to reproduce nuclear matter bulk properties, as the binding energy at the correct saturation density and incompressibility as well as ground state properties of some nuclei [23—25]. The present work provides the necessary formalism for the investigation of how the non extensive statistics affects stellar matter. In this limit Eq. There is an intrinsic gain in the knowledge of the non extensive partition function, since it is closer to the methods of Statistical Mechanics, however in order to go deeper into more fundamental aspects one needs knowledge about nonperturbative QCD that are not available at present.

The integrand in Eq. From Eq. Thus we conclude that the occupation number derived from the partition function defined in Eq.

This result is identical to the CMP entropy defined in Ref. Finally, for the sake of completeness, we show also the result for the average energy. These terms are consequence of the discontinuity in the integrand of Eq. Note that we are considering the same definitions for these functions as the ones used in many previous references, in particular [26, 27].

The existence of this discontinuity was already pointed out and discussed in [27] see Fig. This means that the discontinuity will lead to some contribution proportional to it when deriving with respect to the chemical potential, which should be added to the contribution from the first derivative of the integrand, as it is written in Eq.

It would be possible to get a continuous integrand in Eq. See e. One way to avoid the explicit appearance of the p-independent terms in Eqs. The entropy is then obtained by deriving the partition function with respect to temperature, as in Eq. The entropy of Eq.

On the other hand, it is worth mentioning that the entropy of [21] Eqs. This means that these authors are not consistent with each other.

H15LD 0. H21L s GeV3 D 0. With these results we conclude that the partition function in Eq. Therefore definition given here is in accordance with the non extensive self-consistent thermodynamics. In the following we explore some of the features of the thermodynamical systems described by the partition function written in Eq.

There is an intrinsic gain in the knowledge of the non extensive partition function, since it is closer to the methods of Statistical Mechanics. However, in order to go deeper into more fundamental aspects one needs knowledge about nonperturbative QCD that are not available at present.

In fact, due to the transition from confined to deconfined regimes, the hadronic matter can be found only below the phase transition line. There are different proposals for the conditions determining the transition line. This result was obtained through a sys- tematic analysis of particle yields from HEP experiments. In [30] the transition line was obtained in terms of the total baryon density with the help of the hadron resonance gas model.

In [31], the freeze-out condition was determined from an interpolation between a resonance gas used at low densities and repulsive nucleonic matter at low temperatures. In [32], the author proposed that the chemical freeze-out of hadrons in heavy-ion collisions is characterized by the entropy density and its value was taken from Lattice QCD LQCD calculations at zero chemical potential. In an interesting and more recent analysis [33], higher order multiplicity moments obtained with the hadron resonance gas were used in the calculation of the standard deviation, the variance or susceptibility , the skewness and the kurtosis, quantities which are related to the cumulants [34], that can be experimentally determined and are also of interest in LQCD calculations.

The chemical freeze-out curve was then described in terms of the susceptibility of the system. However we present below some evidences that this hypothesis is correct. In the following we assume that the entropic index, qo , is a fixed property of the hadronic matter with its value determined in the analysis of pT -distributions and in the study of the hadronic mass spectrum in Ref. The system of interest here is a gas composed by different hadronic species in thermodynamical and chemical equilibrium.

The lowest-lying hadrons considered in our calculations are taken from the Particle Data Group [35], and some of them are presented in Tables I and II. The computation will be performed by restricting the ensemble summation in Eq. After that, the effect of a nonzero value of the chemical potential for pions will be explored as well.

In addition, one can observe in Fig. The phase transition line obtained with the energy per particle condition, as described above, is reported in Fig. For the sake of comparison the transition line obtained through the entropy condition was calculated, and its results are shown in Fig. We observe that both conditions lead to transition lines that are in agreement with the available experimental data. We observe in Fig. List of the lowest-lying mesons used in Eq.

We include some of their properties: mass, strangeness S and degeneracy g. The dots indicate that heavier mesons are included in the computation, although they are not explicitly shown in this table due to lack of space. List of the lowest-lying baryons used in Eq. This constant factor is of the order of 5 when using the entropy criterium. In the following we will restrict our discussion to the results obtained with the energy per particle hypothesis.

The curves in Fig. This increase is displayed in Fig. A similar behavior is observed in Boltzmann-Gibbs statistics. Dashed green line corresponds to the density of protons, while continuous red line is the density of neutrons. The transition line determines the region where the confined states exist below the line , and the region where one expects to find the quark-gluon plasma above the line.

Left: Pressure as a function of temperature. In Fig. We show in Fig. From Figs. Left: Result using Boltzmann-Gibbs statistics. Left: Boltzmann-Gibbs statistics. Up to now we have studied the equation of state including the spectrum of hadrons in Table I and II. It would be interesting to analyze also the case in which only protons, neutrons and possibly pions contribute to the equation of state, all of them with nonzero chemical potential. These values are relevant for the study of protoneutron stars [42].

The phase transition lines in Boltzmann-Gibbs and in Tsallis statistics in the regime of high baryonic chemical potential are plotted in Fig. The effect of pions is to increase the values for the pressure with respect to the case with only protons and neutrons. When considering a nonzero value for the pion chemical potential this leads to a noticeable effect on the EOS, as it becomes harder either in Boltzmann-Gibbs or in Tsallis statistics, see Fig.

In the limit of high energies, the partition function is in accordance with that proposed in Ref. The same as in Fig. See Figs. Some thermodynamical functions are derived from the partition function for hadronic systems with different values of chemical potentials.

Particularly we analyze how pressure and energy densities vary when the entropic index or the chemical potentials vary, and we obtain the chemical freeze-out line by using two different hypotheses. A discussion about a discontinuity observed in the first derivatives of the partition function is done for the first time. The results presented in this work can be applied to stellar matter, where high pressures are necessary to compensate for the gravitational force so that protoneutron star stability and correct macroscopic properties are attained.

This study is performed in other work [42], and the results compared with the ones existing in the literature [43]. The research of E. Tsallis, J. Bediaga, E. Curado e J. Hagedorn, Nuovo Cimento Suppl.

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