Expansion against constant pressure

EXAMPLE 6.1 Calculating the work done by expansion against constant pressure... 

The fact that Pext is constant enables it to be taken outside the integral sign (Frame 2). Thus, in general for expansion against constant pressure, P ... 

SOLUTION For expansion against constant external pressure we use Eq. 3 and for reversible, isothermal expansion we use Eq. 4 ... 

Which process will yield more work (1) expansion of a gas confined by piston against constant pressure or (2) reversible expansion of a gas confined by a piston ... 

The system expands as the temperature increases at constant pressure. Work is needed against the molecules and the atmosphere as they move apart. In the critical region the coefficient of thermal expansion at constant pressure is large and thus the effect on Cp is also large. 

In a batch process at constant pressure, where work is done only in expansion against the pressure, the heat absorbed by the system is the gain in enthalpy,... 

At the instant a pressure vessel ruptures, pressure at the contact surface is given by Eq. (6.3.22). The further development of pressure at the contact surface can only be evaluated numerically. However, the actual p-V process can be adequately approximated by the dashed curve in Figure 6.12. In this process, the constant-pressure segment represents irreversible expansion against an equilibrium counterpressure P3 until the gas reaches a volume V3. This is followed by an isentropic expansion to the end-state pressure Pq. For this process, the point (p, V3) is not on the isentrope which emanates from point (p, V,), since the first phase of the expansion process is irreversible. Adamczyk calculates point (p, V3) from the conservation of energy law and finds... 

In the combustion reaction as carried out in the calorimeter of Figure 7-2, the volume of the system is kept constant and pressure may change because the reaction chamber is sealed. In the laboratory experiments you have conducted, you kept the pressure constant by leaving the system open to the surroundings. In such an experiment, the volume may change. There is a small difference between these two types of measurements. The difference arises from the energy used when a system expands against the pressure of the atmosphere. In a constant volume calorimeter, there is no such expansion hence, this contribution to the reaction heat is not present. Experiments show that this difference is usually small. However, the symbol AH represents the heat effect that accompanies a chemical reaction carried out at constant pressure—the condition we usually have when the reaction occurs in an open beaker. 

We can relate pressure to the work of expansion against a constant pressure by using the fact that pressure is the force divided by the area to which it is applied P = PI A (Section 4.2). Therefore, the force opposing expansion is the product of the pressure acting on the outside of the piston, Pex, and the area of the piston, A (P = P(XA). The work needed to drive the piston out through a distance d is therefore... 

The work done by any system on its surroundings during expansion against a constant pressure is calculated from Eq. 3 for a reversible, isothermal expansion of an ideal gas, the work is calculated from Eq. 4. A reversible process is a process that can be reversed by an infinitesimal change in a variable. 

Expansion work against constant external pressure ... 

So far, we have been looking at systems for which the external pressure was constant. Now let s consider the case of a gas that expands against a changing external pressure. In particular, we consider the very important case of reversible, isothermal expansion of an ideal gas. The term reversible, as we shall see in more detail shortly, means that the external pressure is matched to the pressure of the gas at every stage of the expansion. The term isothermal means that the expansion takes place at constant temperature. In an isothermal expansion, the pressure of the gas falls as it expands so to achieve reversible, isothermal expansion, the external pressure must also be gradually reduced in step with the change in volume (Fig. 6.9). To calculate the work, we have to take into account the gradual reduction in external pressure. 

P) and volume (V) of the system. AH is the amount of heat absorbed from the surroundings if a reaction occurs at constant pressure and no work is done other than the work of expansion or contraction of the system. (The work done when a system expands by AV against a constant pressure P is P AV. This type of work is generally not very useful in biochemical systems.) In most biochemical reactions, little change occurs in either pressure or volume, so the difference between AH and AE is relatively small. 

How big a difference is there between AE, the heat flow at constant volume, and AEf, the heat flow at constant pressure Let s look again at the reaction of propane, C3H8, with oxygen as an example. When the reaction is carried out at constant volume, no PV work is possible and all the energy is released as heat AE = —2045 kj. When the same reaction is carried out at constant pressure, however, only 2043 kj of heat is released (AH = —2043 kj). The difference, 2 kj, occurs because at constant pressure, a small amount of expansion work is done against the atmosphere as 6 mol of gaseous reactants are converted into 7 mol of gaseous products. 

If the expansion is carried out stepwise, first against the constant pressure P3, then against the constant pressure jv and finally against the constant pressure P2, as illustrated in Figure 3.2, then the amount of work done on... 

Since PAV represents the quantity of work done by gas on expansion against the external pressure, therefore, - AG gives the maximum work which can be obtained from the system other than that due to change of volume, at constant temperature and pressure. The work other than that due to change of volume is called the net work. Hence,... 

When energy, e.g. in the form of heat, is supplied to the system under constant pressure, only a fraction of this energy serves to increase the internal energy of the system U. The remainder of the energy goes for expansion work (volumetric work) against the external pressure. The sum of V and the volumetric expansion work p-V s the enthalpy H. With entropy of a system defined as the ratio of the amount of heat q and temperature, S = q/T, the two quantities A = U - T S and G = H - T S are thus defined. 

In the second case, after half the initial force has been removed, the gas undergoes a sudden expansion against a constant force equivalent to a pressure of 7 bar. Eventually the system returns to an equilibrium condition identical with the final state attained in the reversible process. Thus A V is the same as before, and the net work accomplished equals the equivalent external pressure times the volume change, or... 

If the change described in Exercise 1.19.13 is carried out in two stages, (a) an adiabatic compression to 50°C using a constant external pressure of 18.1 atm, and (b) an isothermal expansion against a constant external pressure of 2.0 atm, find AE, AH, W, and Q for each stage and also for the overall process. Compare the overall values with those of Exercise 1.19.14. Why is Q AH in (a), even though Pex was constant ... 

A certain gas obeys the equation of state P(V -nb) - nRT and has a constant volume heat capacity, Cv, which is independent of temperature. The parameter b is a constant. For 1 mol, find W, AE, Q, and AH for the following processes (a) Isothermal reversible expansion. (b) Isobaric reversible expansion. (c) Isochoric reversible process, (d) Adiabatic reversible expansion in terms of Tlf Vlt V2, Cp, and Cv subscripts of 1 and 2 denote initial and final states, respectively. (c) Adiabatic irreversible expansion against a constant external pressure P2, in terms of Plf P2, Tj, and 7 = (Cp/Cy). 

Enthalpy change, A//, is equal to the heat involved in a process when the process is done under a constant pressure and involves no work except perhaps expansion (or contraction) against the atmosphere. When these conditions are not met. A// is a more fundamental quantity than heat. For example. A// is a state function, which means that the change in its value is independent of the path in going from the initial state to the final state. Another example of a state function is change in volume. For example, if a gas starts out occupying 2.5 L and finally occupies 4.5 L, AF = 2.0 L no matter if the gas is first expanded to 8.6 L, then contracted to 3.7 L, then expanded to 4.5 L, or if some other path were followed. (Heat, in contrast, does depend on the path, except when q = A//.) The A// value is important in both theoretical and practical terms in chemistry. 

If a system expands in volume by an amount AV(m- ) against a constant restraining pressure P(N/m ), a quantity PAV (J) of energy is transferred as expansion work from the system to its surroundings. Suppose that the following four conditions are satisfied for a closed system (a) the system expands against a constant pressure (so that Ap = 0) (b) A k == 0 (c) A p = 0 and (d) the only work done by or on the system is expansion work. Prove that under these conditions, the energy balance simplifies ioQ = A//. 

---