Work of Expansion

In this frame we bring together the properties of ideal gases (See Sections 4.1 and 4.2 Frame 4) and the specific calculation of work done when a gas is expanded (Frame 7). 

If one suddenly opens the tap (valve) on a cylinder containing a gas confined under a pressure Pi (much greater than atmospheric pressure, Patm (i.e. P, 3 Patm)) and allows it to escape by into the atmosphere this process will continue until the pressures are equalised and the final pressure Pf = Patm. The expansion (leaving aside all discussion of throttle effects at the valve, gas/air mixing, friction effects etc.) takes place rapidly - and under non-equilibrium conditions - usually at constant temperature, T (= ambient) and is a spontaneous process. Since this process is not at equilibrium and hence is not reversible, we refer to it as being an irreversible process. 

The work done on expansion of the gas from initial volume, Fj, pressure, Pi to a final volume Vf, pressure, Pf is calculated using equation (8.5), Frame 8, recognising that P is a constant throughout (i.e. the process is isobaric). Pi (at the instant the tap is opened and the expansion begins) is equal to Pf = Pext = Patm (on opening the tap the gas experiences atmospheric pressure). Accordingly  

If the above expansion is carried out adiabatically (i.e. so rapidly that no heat can enter or leave the system, thus q = 0) then since  

This process is not one that a real system actually can undergo. It is a theoretical process. 

Intlal pressure P. suddenly dhanges to P, (example as when opening gas cylinder tap, gas confined at pressure P, then suddenly ecperiences pressure P. = P.. = 1bar 

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Many systems involve only work of expansion or compression of the system boundaries. For such systems the first law is written for unit mass of fluid as the basis ... 

The unit in which the work of expansion of a fluid is expressed will depend on the units adopted for p and v. Its dimensions... 

Figure 2.3 Work of expansion for an isobaric process. In the expansion. pcxl is constant and equal to p, the pressure of the fluid, unless a mechanical constraint prevents the two pressures from being equal.

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 maximum potential power of an explosive can be calculated, or it can be measured by techniques such as those developed by Cook. A typical method consists of firing the explosive under water and measuring the energy liberated in the various forms, such as shock wave in the water, the work of expansion of the gas bubble, etc. These figures have limited practical value as the methods of application of explosives are of low and variable efficiency. A more practical measurement of strength can be obtained by the measurement of cratering efficiency. This, again, demands considerable expense and also requires the availability of uniform rock. 

If the volume of the system is kept constant when the heat is added to a system then no work is done by the system. Thus the heat absorbed by the system is used up completely to increase the internal energy of the system. Again if the pressure of the system is kept constant when the heat is supplied to the system then some work of expansion is also done by the system in addition to the increase in internal energy. Thus if at constant pressure, the temperature of the system is to be raised through the same value as at constant volume, then some extra heat is required for doing the work of expansion. Hence Cp >C,... 

This quantity AG is equal lo the maximum net work available (i.e., work, other than work of expansion, in a reversible process) for a given change in slate under constant temperature and 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. 

We can write the differential of the energy function in terms of the differentials of the independent variables that we choose to define the state. We will find in Chapter 5 that only two independent variables need to be used if the system is closed and only the work of expansion is involved. The two most convenient variables to use here are the temperature and volume. The differential of the energy function in terms of T and V is given by... 

The term d W is the maximum work done by the surroundings, and thus the negative of the maximum work done by the system, excluding the work of expansion or compression. We emphasize that such an interpretation is valid only for an isothermal and isopiestic reversible process. 

Equation (4.17) is applicable to quasistatic processes for the work of expansion and compression. On the introduction of the second law we have four equations for closed systems... 

In the remainder of this book we consider all work terms other than the work of expansion and compression to be zero, except in those specific cases where such terms are definitely considered. This means that the generalized coordinates that appear as differentials in the expression for such work terms are all considered to be constant. 

In order to save space, the subscript n in a derivative indicates that the number of moles of all components are kept constant except in the case when the variation is one of the components. In that case it indicates that the number of moles of all of the components are kept constant except the one that is being varied. Also, it must be remembered that all work terms other than the work of expansion or compression are zero, even though the notation is not made in general practice. 

We may classify all calorimeters into two groups when we limit the processes to those that involve only the work of expansion or compression those that operate at constant volume and those that operate at constant pressure. The application of Equation (9.1) to constant-volume calorimeters shows that the heat absorbed by the system equals the change of energy of the system for the change of state that takes place in the system. Similarly, the heat absorbed by the system in constant-pressure calorimeters is equal to the change of enthalpy for the change of state that takes place in the system according to Equation (9.2). 

The type of work that we will deal with most often in this book is work of expansion and contraction, which we will call PVwork. Usually, the expansion or contraction is against the pressure of the atmosphere. In cases in which other types of work are involved, such as the work required to stretch an object or increase its surface area or the work of electrochemical cells or driving chemical reactions, we will usually designate these as vvoth or 8vvoth. We then have... 

This means the following The change in free energy in a reaction is equal to the total reversible work obtainable from the reaction (this includes all kinds of work, i.e., gravitational, electrical, surface, etc., and also the work of expansion) diminished by the work of expansion, PAV. Hence,... 

We shall discuss adoption of a convention for the sign for work of expansion -(Frames 7, 9, 14 and 15) and use it when we discuss in more detail the gas expansion processes (Frame 9). Also (FIRST LAW OF THERMODYNAMICS - see Frames 2, 8) the internal energy change, A U for the overall process in Figure 1.1 (i.e. gas at Vf and 7j -> gas at Vf and Tf) (being a state function) is identical for both paths between the SAME initial and final states (and so is route independent). 

Figure 1.1 Comparison of path dependent functions (q and w) and path independent change (AU) during the expansion of a gas from (V, T,) to (Vf, 7)) via two different states (V, T() (path 1) and (Vf, T) (path 2). 1/ and T represent the volume and temperature of the gas. q and w represent the heat absorbed by the gas and the work done by the gas on the surroundings in expanding against the external pressure, P. Wf and W2 are both negative (using the convention discussed in Frame 7) since work of expansion is expended by the gas and lost from the (gas) system.

This mechanical work of expansion is often referred to as PV work. 

In other cases (Figure 8.2c) heat <73 (<73 > q > q2) can be supplied, part of which can be used to increase the internal energy of the system but some of which can be used by the water to increase its volume (by expanding). In this expansion process energy is needed to force back the external pressure imposed on its surface by the atmosphere thus doing work of expansion against the surroundings (= —wf). For this case ... 

Work can take forms other than that of mechanical (PV) work of expansion and contraction. We can experience electrical, gravitational, magnetic, surface and other forms of work (= w ). Most generally then, we can state the First Law of Thermodynamics by means of the following equation ... 

In order to describe the heat changes, qP which take place in a system (where the external pressure P is constant - most usually being equal to Patm (= 1 bar)) we need to rearrange equation (10.1), since in this system work of expansion (leading to a change of volume) may now be permitted, so that ... 

This enthalpy change is used both to raise the temperature and to provide work of expansion of the material as its temperature increases, and so the heat capacity can be defined as ... 

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