Entropy change with pressure

EXAMPLE 9.4 Entropy change with pressure. Consider the dependence of entropy on pressure, 5(p), at constant temperature ... 

As opposed to enthalpy, the entropy change with pressure of an ideal gas is nonzero. 

Since the lattice is fixed its volume does not change with pressure, and the Gibbs and Helmholtz energies of the system are the same. Adding the energy part from eq.(12.20 ) and eq.(12.21) and the entropy part gives ... 

In its most general form, the fullerene synthesis could be treated as a complex kinetic scheme described by a huge number of kinetic differential equations. The equilibrium composition comes as the limiting case for infinite time. If we treat the problem from a thermodynamic point of view, we should realize that the conventional standard pressure of 1 atm is considerably different from the actual fullerene synthesis conditions. We should expect lower cluster pressures in the carbon-arc synthesis. The actual entropy and Gibbs free energy change with pressure as can be demonstrated [208-212] on the Cgo and C70 cases based on computed or observed [213] data. For example, the equilibrium constant Xgo/yo for an interconversion between the two clusters, expressed in partial pressures p, offers a deeper insight into the problem [208-212] ... 

In chapter 4.7, it was assumed that the influence of pressure on the entropy content of sohd and liquid substances can normally be disregarded. For ideal gases, the entropy change with the pressure p is determined by eqn. (4.22). We shall now estimate this assumption based on Maxwell s relations. 

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 ... 

Equation (3.16) shows that the force required to stretch a sample can be broken into two contributions one that measures how the enthalpy of the sample changes with elongation and one which measures the same effect on entropy. The pressure of a system also reflects two parallel contributions, except that the coefficients are associated with volume changes. It will help to pursue the analogy with a gas a bit further. The internal energy of an ideal gas is independent of volume The molecules are noninteracting so it makes no difference how far apart they are. Therefore, for an ideal gas (3U/3V)j = 0 and the thermodynamic equation of state becomes... 

The effect of compressibility is important in high mach number machines. Mach number is the ratio of velocity to the acoustic speed of a gas at a given temperature M = Vja. Acoustic speed is defined as the ratio change in pressure of the gas with respect to its density if the entropy is held constant ... 

Figure 2.13 Mixing of ideal gas A with ideal gas B at constant temperature and constant total pressure. The entropy change AS is given by equation (2.78).

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. 

Coefficient of Expansion The change in entropy with pressure is related to the coefficient of expansion by... 

EXAMPLE 7.4 Sample exercise Calculating the change in entropy with pressure... 

Assuming that the heat capacity of an ideal gas is independent of temperature, calculate the entropy change associated with lowering the temperature of 2.92 mol of ideal gas atoms from 107.35°C to —52.39°C at (a) constant pressure and (b) constant volume. 

As a result, there will be a continuous change in G as the transition of one phase into another takes place. However, for some phase transitions (known as first-order transitions), it is found that there is a discontinuity in the first derivative of G with respect to pressure or temperature. It can be shown that the partial derivative of G with pressure is the equal to volume, and the derivative with respect to temperature is equal to entropy. Therefore, we can express these relationships as follows ... 

The variation of the phase transition temperature with pressure can be calculated from the knowledge of the volume and enthalpy change of the transition. Most often both the entropy and volume changes are positive and the transition temperature increases with pressure. In other cases, notably melting of ice, the density of the liquid phase is larger than of the solid, and the transition temperature decreases... 

Because the entropy change for the H2/O2 reaction is negative, the reversible potential of the H2/O2 fuel cell decreases with an increase in temperature by 0.84 mV/°C (assuming reaction product is liquid water). For the same reaction, the volume change is negative therefore, the reversible potential increases with an increase in pressure. 

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... 

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