Work, pressure-volume

The kind of work most frequently associated with chemical change occurs when the volume of the system changes owing to the disappearance or formation of gaseous substances. This is sometimes called expansion work orPF-work, and it can most easily be understood by reference to the simplest form of matter we can deal with, the ideal gas. 

An early statement of energy conservation by Rene Descartes (1596-1650) explained that in creating the world, God established vortices , states of motion that could interact and be transformed, but which in their totality would endure in perpetuity. 

When dealing with a gas, it is convenient to think in terms of the more relevant quantities pressure and volume rather than force and distance. We can accomplish this by multiplying the second term by A/A which of course leaves it unchanged  

By grouping the terms differently, but still not changing anything, we obtain 

Since pressure is force per unit area and the product of the length A and the area has the dimensions of volume, this expression becomes 

To better understand the significmce of enthcdpy, recall from Equation 5.5 that AE involves not only the heat q added to or removed from the system but cdso the work w done by or on the system. Most commonly, the only kind of work produced by chemical or physi-Ccd changes open to the atmosphere is the mechcuiiccd work cissociated with a chcuige in voliune. For example, when the reaction of zinc metal with hydrochloric acid solution 

If the amount of zinc used in the reaction is increased, will more work be done by the system Is there additional information you need to answer this question  

Increase in volume means system does pressure-volume work 

The work involved in the expansion or compression of gases is called pressure-volume work P-V work). When pressure is constant in a process, as in our preceding example, the sign and magnitude of the pressure-volume work are given by 

The units of work obtained by using Equation 5.8 will be those of pressure (usually atm) multiplied by those of volume (usually L). To express the work in the more familiar unit of joules, we use the conversion factor 1 L-atm = 101.3 J. 

Let us begin by considering the process of gas compression in a cylinder of cross-sectional area A, with external force Fext driving the piston downward through a differential distance dr, as shown in the following diagram  

Since the force and displacement are parallel, we can write the general differential work expression (2.60) for PV work in the simpler form 

By dividing one factor and multiplying the other by the cross-sectional area A, we can rewrite (3.8) as 

Note that the sign in (3.10) conforms to the acquisitive convention if the gas (system) is compressed ( dV 0), work is performed on the gas ( dwPV 0), whereas for gas expansion (dV 0), the gas performs useful work on the surroundings (dwPV 0). 

For a general process A — B, the PV work is evaluated as the path integral 

Let us take nii as the initial mass on top of the piston. Am is the amount of mass suddenly removed, and so the final mass is m2 = mj - Am. The initial and final forces due to the mass are 

A cylindrical piston system. The volume of the gas inside the cylinder is the cross-sectional area of the cylinder. A, times the height, h. The mass, m, is placed on top of the piston in increments, giving rise to a downward force due to gravity, mg. (The piston is attached to the bottom mass and its mass is included in the value m ) 

This is simply the change in the gravitational potential energy of the mass, m2, that is on top of the piston as the gas expands. Since m2g is the external force, P, and since F = PA, 

This says that the work expended in the course of the gas expansion is the product of the change in volume and the external pressure. The external pressure is constant in this expansion because the mass was removed instantaneously. If the change in mass were to decrease, the change in volume would decrease as well. Thus, we can say that in the limit of very small changes, the differential in the work is the external pressure times the differential in the volume. 

Equation 3.1 uses the symbol 8w instead of dxv for an important mathematical reason that we next consider. 

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In this and nearly all subsequent sections, the work V>w will be restricted to pressure-volume work, -p dF, and the fact that the heat Dq may in some cases be electrical work will be ignored. 

In the example of pressure-volume work in die previous section, the adiabatic reversible process consisted simply of the sufficiently slow motion of an adiabatic wall as a result of an infinitesimal pressure difference. The work done on the system during an infinitesimal reversible change in volume is then -pdVand one can write equation (A2.1.11) in the fomi... 

Thennal equilibrium means free transfer (exchange) of energy in the fonn of heat, mechanical (liydrostatic) equilibrium means free transfer of energy in the fonn of pressure-volume work, and material equilibrium means free transfer... 

If there is no volume change (dV= 0), then obviously there is no pressure-volume work done (du = 0) irrespective of the pressure, and it follows from equation (A2.1.10) that the change in energy is due entirely to the heat absorbed, which can be designated as qy. 

Note that in this special case, the heat absorbed directly measures a state fiinction. One still has to consider how this constant-volume heat is measured, perhaps by an electric heater , but then is this not really work Conventionally, however, if work is restricted to pressure-volume work, any remaining contribution to the energy transfers can be called heat . 

If there are other kinds of work, similar expressions apply. For example, with electromagnetic work (equation (A2.1.8)1 instead of pressure-volume work, one can write for the Helmholtz free energy... 

A consistent sign convention has been applied to the pressure-volume work term A positive dV corresponds to an expanded system, and work is done by the system to push back the surrounding atmosphere. 

P = 1 atm Figure 8.11 Pressure-volume work. When the reaction... 

Students often ask, What is enthalpy The answer is simple. Enthalpy is a mathematical function defined in terms of fundamental thermodynamic properties as H = U+pV. This combination occurs frequently in thermodynamic equations and it is convenient to write it as a single symbol. We will show later that it does have the useful property that in a constant pressure process in which only pressure-volume work is involved, the change in enthalpy AH is equal to the heat q that flows in or out of a system during a thermodynamic process. This equality is convenient since it provides a way to calculate q. Heat flow is not a state function and is often not easy to calculate. In the next chapter, we will make calculations that demonstrate this path dependence. On the other hand, since H is a function of extensive state variables it must also be an extensive state variable, and dH = 0. As a result, AH is the same regardless of the path or series of steps followed in getting from the initial to final state and... 

The combination of fundamental variables in equation (l.23) that leads to the variable we call G turns out to be very useful. We will see later that AG for a reversible constant temperature and pressure process is equal to any work other than pressure-volume work that occurs in the process. When only pressure-volume work occurs in a reversible process at constant temperature and pressure, AG = 0. Thus AG provides a criterion for determining if a process is reversible. Again, since G is a combination of extensive state functions... 

Equation (2.7) can be applied to processes involving many forms of work. Our present interest is in calculating pressure-volume work, that is, the work done in changing the volume of a system by an amount d V against an external pressure pext. The process is shown in Figure 2.1. A fluid of volume V is confined in a cylinder. The fluid exerts a pressure p on a piston this pressure is balanced by an external pressure pext so that the piston does not move. The external pressure is then decreased by an amount dp.c This causes the piston to move by an amount dx until the internal pressure decreases by dp and equilibrium is again established. The work accomplished by this process is given by equation (2.8),... 

Figure 2.1 Decreasing the external pressure by dp causes the piston to move a distance dx. This increase in volume reduces the internal pressure by dp and re-establishes mechanical equilibrium. The process produces pressure-volume work.

Since q2>0 and q < 0, and q2 is greater in magnitude than qu the cyclic engine produces a net amount of (negative) work. From our earlier discussion of pressure-volume work, we know that the area enclosed by the cycle in a p-V plot gives the magnitude of this work.p... 

We have previously shown that the Pfaff differential <5inexact differential. It is easy to show that division of equation (2.43) by the absolute temperature T yields an exact differential expression. The division gives... 

These equations can be used to derive the four fundamental equations of Gibbs and then the 50,000,000 equations alluded to in Chapter 1 that relate p, T, V, U, S, H, A, and G. We should keep in mind that these equations apply to a reversible process involving pressure-volume work only. This limitation does not restrict their usefulness, however. Since all of the thermodynamic variables are state functions, calculation of AZ (Z is any of these variables) by a reversible path between two states gives the same value as would be obtained for all other paths between those states. When other forms of work are involved, additions can be made to the equations to account for the additional work. The... 

A reversible isothermal expansion of the ideal gas is made from an initial volume V to a volume Vz at an absolute (ideal gas) temperature 73. The amount of pressure-volume work in done by the system is obtained by substituting into Equation (2.16). The result is... 

Thus, in an isothermal reversible process, dA equals the reversible work. Note that <5vr in equation (3.92) is the total work. It includes pressure-volume work and any other forms, if present."1... 

A relationship between dG and work can also be obtained. To do so we divide the work for a reversible process into pressure-volume work (—p dV) and 8w, the work other than pressure-volume work that may occur, and write... 

Thus, in a reversible process that is both isothermal and isobaric, dG equals the work other than pressure-volume work that occurs in the process." Equation (3.96) is important in chemistry, since chemical processes such as chemical reactions or phase changes, occur at constant temperature and constant pressure. Equation (3.96) enables one to calculate work, other than pressure-volume work, for these processes. Conversely, it provides a method for incorporating the variables used to calculate these forms of work into the thermodynamic equations. 

If we restrict the work in our process to pressure-volume work, then... 

The preceding calculations can also be performed for finite cavity sizes. For this case, there are some additional sources of small amounts of energy associated with cavity formation arising from surface tension, pressure-volume work, and electrostriction. Because of the Franck-Condon principle these do not affect the transition energy, but they have some influence on the heat of solvation. Jortner s (1964) results are summarized as follows ... 

When a fully inflated car tyre is allowed to deflate, the air streaming through the nozzle is cold to the touch. The pressure of the air within the tyre is fairly high, so opening the tyre valve allows it to leave the tyre rapidly - the air movement may even cause a breeze. We could feel a jet of cold air on our face if we were close enough. As it leaves the tyre, this jet of air pushes away atmospheric air, which requires an effort. We say that work is performed. (It is a form of pressure-volume work, and will be discussed in more depth later, in Section 3.2.)... 

We often calculate a volume of AU but cite the answer after adjusting for pressure-volume work see p. 102. 

The magnitude of this pressure-volume work is w, and is expressed by... 

Most chemists perform experiments in which the contents of our beaker, flask or apparatus are open to the air - obvious examples include titrations and refluxes, as well as the kinetic and electrochemical systems we consider in later chapters. The pressure is the air pressure (usually pe), which does not change, so any pressure-volume work is the work necessary to push back the atmosphere. For most purposes, we can say w = pAV. 

But not all of the heater s energy q goes into raising U. We need some of it to perform pressure-volume work, since the vapour formed on boiling works to push back the external atmosphere. The difference between the internal energy U and the available energy (the enthalpy) is given by... 

The change in enthalpy A H during a thermodynamic process is defined in terms of internal energy and pressure-volume work by... 

Ablution) is sometimes called heat of solution, particularly in older books. The word heat here can mislead, and tempts us to ignore the possibility of pressure-volume work. 

In the absence of any pressure-volume work, the value of AG(ceii) is equal to the work needed to transfer charge from the negative end of the cell to the positive. In practice, AG(ceii) equates to the amount of charge passed, i.e. the number of charged particles multiplied by the magnitude of that charge. 

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