Energy, Entropy, and Temperature

If we know the probability that a system exists in a certain state i is the value P this value can be used in place of the number of configurations in Equation 1.6. P and C must be proportional to each other, and so it is sufficient to know P even if we do not know the absolute numbers of configurations, C. Therefore, 

Dividing by k and taking the exponential of this equation gives 

The probability of being in state 2 is related to the probability of being in state 1 via an exponential in the entropy difference, AS. 

The entropy of the molecule does not change, assuming that all the states are nondegenerate, because there is only one configuration for each state. That is, AS(A) = 0. Therefore, the entropy change for the system is the same as AS(B), which is AE(B)/T according to our postulate. 

This can be related to the probabilities that the system exists in the two states being considered. 

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As to free energy, below we shall use the Helmholtz free energy F = U — TS (U, S and T are total energy, entropy and temperature, respectively) that is more appropriate for discussion of the systems in terms of temperature and volume V (or density p) at constant pressure p. In a more general case, the thermodynamic potential (or Gibbs free energy) (b = F + pV appears to be more suitable for an expansion, e.g. when varying pressure p. 

The relationship between energy, entropy and temperature for reversible processes is best described using the first and second law of thermodynamics. The first law relates the change in internal mechanical energy, dU, and the change in internal thermal energy (heat), dQ, to the work done on the system by external forces, dW, and is given as. 

This method was first applied by McCormick27 and by Bywater and Worsfold11 to the system a-methylstyrene/poly-a-methyl-styrene, and the free energy, entropy and heat of polymerization as well as the ceiling temperature were determined. Similar studies concerned with the system styrene/polystyrene are being carried out in our laboratories. 

With motion along the connodal curve towards the plait point the magnitudes Ui and U2, Si and S2, and ri and r2, approach limits which may be called the energy, entropy, and volume in the critical state. The temperature and pressure similarly tend to limits which may be called the critical temperature and the critical pressure. Hence, in evaporation, the change of volume, the change of. entropy, the external work, and the heat of evaporation per unit mass, all tend to zero as the system approaches the critical state ... 

Table 8 Helmholtz free energy, entropy, and internal energy of spreading and of transition for N-stearoyltyrosine on an aqueous subphase of pH = 6.86 at the transition temperature for each film."... 

The Gibbs free energy (G) is a thermodynamic function that combines the enthalpy, entropy, and temperature ... 

Now recall the reaction between mercury and oxygen. It favours the formation of HgO below about 400°C, but the decomposition of HgO above 400°C. This reaction highlights the importance of temperature to favourable change. Enthalpy, entropy, and temperature are linked in a concept called free energy. 

The relationship among heat capacity, entropy, and temperature in crystalline solids may be understood on the basis of two fundamental concepts the Boltzmann factor and the partition function (or summation over the states, from the German term Zustandsumme). Consider a system in which energy levels Eq,... 

The important parts of Eq. (4.76) are the exponential terms. The first exponential, which contains the entropies associated with potassium ion movement and vacancy formation, respectively, form the temperature-independent contributions to Do, as discussed in the previous section. The second exponential, which contains the enthalpies of the two processes and the temperature dependence, form the activation energy, Ea, and temperature dependence of Eq. (4.71). 

Gibbs free energy describes the spontaneity of chemical reactions in terms of enthalpy, entropy, and temperature. Negative values signify a spontaneous reaction, while positive values are nonspontaneous. A free energy of zero denotes equilibrium conditions. 

This expression serves as a precise mathematical definition of temperature. It is interesting to note that temperature, a variable with which we have intuitive and sensory familiarity, is defined based on entropy, one with which we may be less familiar. In fact, we shall see that entropy and temperature are intimately related in the concept of free energy, in which temperature determines the relative importances of energy and entropy in driving thermodynamic processes. 

The work of Carnot, published in 1824, and later the work of Clausius (1850) and Kelvin (1851), advanced the formulation of the properties of entropy and temperature and the second law. Clausius introduced the word entropy in 1865. The first law expresses the qualitative equivalence of heat and work as well as the conservation of energy. The second law is a qualitative statement on the accessibility of energy and the direction of progress of real processes. For example, the efficiency of a reversible engine is a function of temperature only, and efficiency cannot exceed unity. These statements are the results of the first and second laws, and can be used to define an absolute scale of temperature that is independent of ary material properties used to measure it. A quantitative description of the second law emerges by determining entropy and entropy production in irreversible processes. 

As AG° becomes more negative, K becomes larger a decrease in free energy favors a given reaction. As we saw in Chapter 10, free energy depends on enthalpy, entropy, and temperature. For a process at constant temperature,... 

Relative free energy, entropy, and enthalpy Relative solubility Transition temperature Critical water/solvent activity... 

Thermodynamics is a branch of physics concerned with heat and temperature and their relation to energy and work. It defines macroscopic variables, such as internal energy, entropy, and pressure, that partly describe a body of matter or radiation. It states that the behavior of these variables is subject to general constraints that are common to all materials, not to the peculiar properties of particular materials. 

Note that the volume changes for the last two processes are identical. We note also that for the liquid phases at room temperature k1t is much smaller than 1 atm-1 (e.g., for water at 0°C, kTxT 1 cm3 mol-1, AV 20 cm3 mol-1, and kT / atm w2x 104cm3 mol-1). Similarly, in equation (7.73) k1t < i P-1 (the limit of an ideal-gas phase). Thus, the volume change for the three standard processes is dominated by the terms which originate from the ideal-gas compressibility. Because of this undesirable feature, it is common to abandon these processes when studying the volume of solvation. Almost all researchers who study the solvation phenomena apply one of these standard processes for quantities like the Gibbs energy, entropy and enthalpy of... 

The intensive porperty Temperature is supplemented by a complementary extensive property, entropy. In the case of energy in form of heat it gives the number of degrees of freedom among which the average energy of motion (of the material particles involved), characterized by the temperature, is distributed. Entropy and temperature are complementary state variables. 

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