Entropy at constant volume

The increase of energy per unit increase of entropy at constant volume is measured by the absolute temperature. 

Figure 5.3. The energy as a function of the entropy at constant volume.

This shows that the entropy at constant volume and composition rises with temperature, as is intuitively evident also, it shows that the heat capacity at constant volume and composition cannot be negative. 

If equation (20.26) is differentiated with respect to volume, at constant entropy, and (20.27) with respect to entropy, at constant volume, and the... 

Abe, A., and Nam, S. Y, PVT smdies on dimer liquid crystals and estimation of transition entropies at constant volume. Macromolecules, 28, 90-95 (1995). 

Orwoll, R. A., Sullivan, V. J., and Campbell, G. C., Thermal pressure coefficients and specific volumes of cyanobiphenyls and their transition entropies at constant volume. Mol. Cryst. Liq. Cryst., 149, 121-140 (1987). 

At constant volumes y" and yt, the state is changed by the adiabatic perfomiance of work (stirring, nibbing, electrical heating ) until the entropy is changed from to S . 

Applied to a two-phase system, this says that the change in pressure with temperature is equal to the change in entropy at constant temperature as the total volume of the system (a + P) is increased, which can only take place if some a is converted to P ... 

Available data on the thermodynamic and transport properties of carbon dioxide have been reviewed and tables compiled giving specific volume, enthalpy, and entropy values for carbon dioxide at temperatures from 255 K to 1088 K and at pressures from atmospheric to 27,600 kPa (4,000 psia). Diagrams of compressibiHty factor, specific heat at constant pressure, specific heat at constant volume, specific heat ratio, velocity of sound in carbon dioxide, viscosity, and thermal conductivity have also been prepared (5). 

Regular Solution Theory. The key assumption in regular-solution theory is that the excess entropy, is zero when mixing occurs at constant volume (3,18). This idea of a regular solution (26) leads to the equations ... 

Since the entropy is the partial differential coefficient of with r constant, or with p constant, with respect to T, the magnitudes and thermodynamic potentials at constant volume and at constant pressure respectively. 

A sample of nitrogen gas of volume 20.0 L at 5.00 kPa is heated from 20.°C to 400.°C at constant volume. What is the change in the entropy of the nitrogen The molar heat capacity of nitrogen at constant volume, CVm, is 20.81 J-K -mol . Assume ideal behavior. 

STRATEGY We expect a positive entropy change because the thermal disorder in a system increases as the temperature is raised. We use Eq. 2, with the heat capacity at constant volume, Cv = nCV m. Find the amount (in moles) of gas molecules by using the ideal gas law, PV = nRT, and the initial conditions remember to express temperature in kelvins. Because the data are liters and kilopascals, use R expressed in those units. As always, avoid rounding errors by delaying the numerical calculation to the last possible stage. 

FIGURE 7.10 More energy levels become accessible in a lx>x of fixed width as the temperature is raised. The change from part (a) to part (b) is a model of the effect of heating an ideal gas at constant volume. The thermally accessible levels are shown by the tinted band. The average energy of the molecules also increases as the temperature is raised that is, both internal energy and entropy increase with temperature. 

In quest of relationships which will allow evaluation of the desired derivatives of the entropy and energy at constant volume, it is convenient to introduce th work function A—E—TS. Proceeding as above... 

For ordinary deformations of rubberlike substances (excluding swelling phenomena to be discussed in the following chapter) it is permissible to assume constant volume, i.e., axayaz = l. The logarithmic term in Eq. (41) then disappears. In the particular case of elongation at constant volume, = = giving for the entropy of deforma-... 

We can rewrite this in a new form at constant volume and entropy ... 

The change in entropy for temperature changes at constant volume are analogous to those at constant pressure except that Cy replaces Cp. Thus, because PdV = 0,... 

Any characteristic of a system is called a property. The essential feature of a property is that it has a unique value when a system is in a particular state. Properties are considered to be either intensive or extensive. Intensive properties are those that are independent of the size of a system, such as temperature T and pressure p. Extensive properties are those that are dependent on the size of a system, such as volume V, internal energy U, and entropy S. Extensive properties per unit mass are called specific properties such as specific volume v, specific internal energy u, and specific entropy. s. Properties can be either measurable such as temperature T, volume V, pressure p, specific heat at constant pressure process Cp, and specific heat at constant volume process c, or non-measurable such as internal energy U and entropy S. A relatively small number of independent properties suffice to fix all other properties and thus the state of the system. If the system is composed of a single phase, free from magnetic, electrical, chemical, and surface effects, the state is fixed when any two independent intensive properties are fixed. 

The standard entropy difference between the reactant(s) of a reaction and the activated complex of the transition state, at the same temperature and pressure. Entropy of activation is symbolized by either A5 or and is equal to (A// - AG )IT where A// is the enthalpy of activation, AG is the Gibbs free energy of activation, and T is the absolute temperature (provided that all rate constants other than first-order are expressed in temperature-independent concentration units such as molarity). Technically, this quantity is the entropy of activation at constant pressure, and from this value, the entropy of activation at constant volume can be deduced. See Transition-State Theory (Thermodynamics) Gibbs Free Energy of Activation Enthalpy of Activation Volume of Activation Entropy and Enthalpy of Activation (Enzymatic)... 

It should also be recalled that dS = Cp/T)dT at constant pressure and dS = (Cy/T)dT at constant volume where S is the entropy. 

A gas obeying the equation of state P(V B) = RT undergoes a change from the initial state Tp Vjto a final state T2. V2- An expression derived for the entropy change of this gas. The variation of heat capacity at constant volume is given by C, = a + bT + cP. 

In practice, then, it is fairly straightforward to convert the potential energy determined from an electronic structure calculation into a wealth of thennodynamic data - all that is required is an optimized structure with its associated vibrational frequencies. Given the many levels of electronic structure theory for which analytic second derivatives are available, it is usually worth the effort required to compute the frequencies and then the thermodynamic variables, especially since experimental data are typically measured in this form. For one such quantity, the absolute entropy 5°, which is computed as the sum of Eqs. (10.13), (10.18), (10.24) (for non-linear molecules), and (10.30), theory and experiment are directly comparable. Hout, Levi, and Hehre (1982) computed absolute entropies at 300 K for a large number of small molecules at the MP2/6-31G(d) level and obtained agreement with experiment within 0.1 e.u. for many cases. Absolute heat capacities at constant volume can also be computed using the thermodynamic definition... 

Calcd. quantities <.r2>0lnl2 = 1.60 or 4.22a Entropy change at constant volume, (ASu)y... 

Wagner (45) also has derived the relationship given by Equation 14, where a is the coefficient of expansion of the metal the first term on the left is —Sff.p, i.e., the partial excess entropy at constant pressure (free surface conditions) and the first term on the right is the corresponding partial excess entropy under constant volume. By combining Equation 9 with Equation 14 and integrating, Oates and Flanagan (47) have obtained Equation 15, where the reference con-... 

A more tractable approach to shock wave propagation in water is that of Kirkwood and coworkers. For details of this rather involved analysis, the reader is referred to Ref 1, pp 29—33 and 104—106. The basic assumptions of this theory are that behind the shock front the entropy is constant, ie, ds=0, and that the conversion of the expl to its products occurs at constant volume. With these assumptions, it is then possible to get approximate analytical solutions of the equation of motion in terms of the enthalpy of the system... 

For a temperature of 298.15 K, a pressure of 1 bar, and 1 mole of H2S, prepare a table of (1) the entropy (J/mol K), and separately the contributions from translation, rotation, each vibrational mode, and from electronically excited levels (2) specific heat at constant volume Cv (J/mol/K), and the separate contributions from each of the types of motions listed in (1) (3) the thermal internal energy E - Eo, and the separate contributions from each type of motion as before (4) the value of the molecular partition function q, and the separate contributions from each of the types of motions listed above (5) the specific heat at constant pressure (J/mol/K) (6) the thermal contribution to the enthalpy H-Ho (J/mol). 

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