Entropy at constant pressure

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-... 

Upon differentiating with respect to pressure, at constant entropy, in the first case, and with respect to entropy, at constant pressure, in the second case, and equating the results, it is seen that... 

AScavj cavity formation entropy at constant pressure... 

Equations (2-28) and (2-29) represent the changes in enthalpy and entropy (at constant pressure) from the base temperature T to any temperature T, In the calculation of these properties, the constant pressure is taken at atmospheric since most Cp data are available at that pressure. The base temperature T can be any number of different points (0 C, 0 K, etc.). It is set to some extent by the availability of Cp data. For example, a T lower than 0°C should not be chosen if specific heat data are not available below that temperature. 

Hence, the change in enthalpy or entropy, at constant pressure, with temperature is independent of the selected base temperature T. ... 

This is an extremely useful relationship, as we no longer need to measure the change in volume with respect to entropy at constant pressure It equals the isentropic change in temperature with respect to pressure. Notice that we have lost any direct relationship to any energy. 

The change in thermal entropy at constant pressure can be written as ... 

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 ... 

Free energy is related to two other energy quantities, the enthalpy (the heat of reaction measured at constant pressure) and the entropy. S. an energy term most simply visualised as a measure of the disorder of the system, the relationship for a reaction taking place under standard conditions being... 

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). 

For sources, units, and remarks, see Table 2-228. v = specific volume, mVkg h = specific enthalpy, kj/kg s = specific entropy, kJ/(kg-K) c = specific beat at constant pressure, kJ/(kg-K) i = viscosity, 10 Pa-s and k = tberni conductivity, VW(m-K). For specific beat ratio, see Table 2-200 for Prandtl number, see Table 2-369. 

The Joule-Brayton (JB) constant pressure closed cycle is the basis of the cyclic gas turbine power plant, with steady flow of air (or gas) through a compressor, heater, turbine, cooler within a closed circuit (Fig. 1.4). The turbine drives the compressor and a generator delivering the electrical power, heat is supplied at a constant pressure and is also rejected at constant pressure. The temperature-entropy diagram for this cycle is also... 

The simplest method to measure gas solubilities is what we will call the stoichiometric technique. It can be done either at constant pressure or with a constant volume of gas. For the constant pressure technique, a given mass of IL is brought into contact with the gas at a fixed pressure. The liquid is stirred vigorously to enhance mass transfer and to allow approach to equilibrium. The total volume of gas delivered to the system (minus the vapor space) is used to determine the solubility. If the experiments are performed at pressures sufficiently high that the ideal gas law does not apply, then accurate equations of state can be employed to convert the volume of gas into moles. For the constant volume technique, a loiown volume of gas is brought into contact with the stirred ionic liquid sample. Once equilibrium is reached, the pressure is noted, and the solubility is determined as before. The effect of temperature (and thus enthalpies and entropies) can be determined by repetition of the experiment at multiple temperatures. 

For points 2-3, there is constant entropy (S) compression for a one pound of air from Pg to P3. From points 3-5 the air cools at constant pressure, and gives up heat, Q, to the intercooler. From points 5-6 the air is compressed at constant S to the final pressure Pg. Note that point Tj = point Tg for constant temperature. For minimum work Tg = T3. Then the heat, Q, equals the Work, Wl of Figure 12-36B. Figure 12-38 is convenient for estimating the moisture condensed from an airstream, as well as establishing the remaining water vapor in the gas-air. 

When a substance is heated at constant pressure without change of phase through a temperature rise dr the heat absorbed is Cp dr, where Cp is the molar heat capacity at constant pressure, and the entropy increase is... 

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. 

E3.7 A block of copper weighing 50 g is placed in 100 g of HiO for a short time. The copper is then removed from the liquid, with no adhering drops of water, and separated from it adiabatically. Temperature equilibrium is then established in both the copper and water. The entire process is carried out adiabatically at constant pressure. The initial temperature of the copper was 373 K and that of the water was 298 K. The final temperature of the copper block was 323 K. Consider the water and the block of copper as an isolated system and assume that the only transfer of heat was between the copper and the water. The specific heat of copper at constant pressure is 0.389 JK. g l and that of water is 4.18 J-K 1-g 1. Calculate the entropy change in the isolated system. 

The entropy of vaporization, ASvap, is the change in entropy per mole of molecules when a substance changes from a liquid into a vapor. The heat required per mole to vaporize the liquid at constant pressure is equal to the enthalpy of vaporization (A//vap, Section 6.11). It then follows from Eq. 1, by setting = AH, that the entropy of vaporization at the normal boiling point is... 

FIGURE 7.11 The experimental determination of entropy, (a) The heat capacity at constant pressure in this instance) of the substance is determined from close to absolute zero up to the temperature of interest, (b) The area under the plot of CP/T against T is determined up to the temperature of interest. 

We can use Eq. 1 to calculate the entropy change of the surroundings, provided that we assume that the surroundings are so large that their temperature and pressure remain constant. If the enthalpy change of the system is AH, then, for heat transfers at constant pressure, gSLlrr = —AH. We can now use Eq. 1 to write... 

This important formula, which can be derived more formally from the laws of thermodynamics, applies when any change takes place at constant pressure and temperature. Notice that, for a given enthalpy change of the system (that is, a given output of heat), the entropy of the surroundings increases more if their temperature is low than if it is high (Fig. 7.16). The explanation is the sneeze in the street analogy mentioned in Section 7.2. Because AH is independent of path, Eq. 10 is applicable whether the process occurs reversibly or irreversibly. 

The entropy change of the surroundings due to a process taking place at constant pressure and temperature is equal to —AH/T, where AH is the change in enthalpy of the system. 

J 8 Estimate the change in entropy of the surroundings due to heat transfer at constant pressure and temperature (Example 7.10). 

Calculate the standard entropy of vaporization of water at 85°C, given that its standard entropy of vaporization at 100.°C is 109.0 J-K -mol 1 and the molar heat capacities at constant pressure of liquid water and water vapor are 75.3 J-K -mol 1 and 33.6 J-K -mol, respectively, in this range. 

Remember that AG is a measure of the overall change in entropy at constant temperature and pressure, AS is the change in entropy of the system and —AH/T is the change in entropy of the surroundings (Section 7.9). 

Second, the process must occur at constant pressure. Then we can relate the enthalpy change for the system to the entropy change for the surroundings. Recall thatZlH equals q when P is constant ... 

Cp is the specific heat at constant pressure, k is the compressibility at constant temperature. The conversion process of a second-order phase transition can extend over a certain temperature range. If it is linked with a change of the structure (which usually is the case), this is a continuous structural change. There is no hysteresis and no metastable phases occur. A transformation that almost proceeds in a second-order manner (very small discontinuity of volume or entropy) is sometimes called weakly first order . 

If an isobaric temperature change is carried out reversibly, the heat exchanged in the process can be substituted into the expression for the entropy change, and the equations at constant pressure when no work is performed other than PV work are... 

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,... 

However, for the two-phase mixture at constant pressure and ternperamre, the change in entropy depends only on the quantity of heat evolved ... 

The formation of water from gaseous hydrogen and oxygen is a spontaneous reaction at room temperature, although its rate may be unobservably small in the absence of a catalyst. At 298.15 K, the heat of the irreversible reaction at constant pressure is — 285,830 J mol . To calculate the entropy change, we must carry out the same transformation reversibly, which can be performed electrochemicaUy with a suitable set of electrodes. Under reversible conditions, the heat of reaction for Equation (6.99) is —48,647 J mol. Hence, for the irreversible or reversible change... 

If we consider an entropy change at constant pressure, then from Equations (6.49) and (4.53),... 

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. 

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