of work done and the distance traveled in climbing from A to B depend upon the path. The work and the distance traveled are therefore path functions.
and work respectively, and negative if energy is lost from the system as heat or work.
Work and heat are path functions, since the amount of work done or heat lost depends not on the initial and final states of the system, but on how that final state is reached. In changing the internal energy of a system, the amount of energy lost as heat or as work, depends upon how efficiently the energy is extracted. Hence some cars travel further on a given amount of petrol than others depending on how efficiently the internal energy of the petrol is harnessed to do work.
Work
There are a limited number of ways in which energy may be exchanged in the form of work. The most commonly encountered of these is pressure-volume or pV work.
Electrical work may also be performed by a system (see Topics E3, E4 and E5), and this may be accounted for by including an appropriate term, but in most cases this may be discounted. When a reaction releases a gas at a constant external pressure, pex work is done in expanding, ‘pushing back’, the surroundings. In this case, the work done is given by:
w=−pex.∆V
and so the change in internal energy ∆U in such a reaction is:
∆U=−pex.∆V+q
If a reaction is allowed to take place in a sealed container at fixed volume then ∆V=0, and so the expression for ∆U reduces to ∆U=q. This is the principle of a bomb calorimeter.
A bomb calorimeter is a robust metal container in which a reaction takes place (often combustion at high oxygen pressure). As the reaction exchanges heat with the surroundings (a water bath, for example), the temperature of the surroundings changes.
Calibration of the bomb using an electrical heater or standard sample allows this temperature rise to be related to the heat output from the reaction and the value of q, and hence ∆U, obtained.
Heat capacity
When energy is put into a system, there is usually a corresponding rise in the temperature of that system. Assuming that the energy is put in only as heat, then the rise in
The first law 39
temperature of a system is proportional to the amount of heat which is input into it, and they are related through the heat capacity, C:
dq=C.dT (infinitesimal change)
or q=C∆T (finite change when C is temperature independent)
The heat capacity of a substance depends upon whether the substance is allowed to expend energy in expansion work or not, and hence there are two possible heat capacities, the constant volume heat capacity, Cv, which is the heat capacity measured at constant volume, and the constant pressure heat capacity, Cp, which is measured at constant pressure. The two are approximately identical for solids and liquids, but for gases they are quite different as energy is expended in volume expansion work. They are related through the formula:
Cp=Cv+nR
Since, at constant volume, the heat supplied is equal to the change in internal energy, δU, it is possible to write:
∂U=Cv ∂T or ∆U=Cv ∆T
when Cv is independent of temperature.
The molar heat capacity, Cm is the heat capacity per mole of substance:
Cm=C/n
The larger the value of Cm the more heat is required to accomplish a given temperature rise.
B2 ENTHALPY
Key Notes
Enthalpy, H, is defined by the relationship H=U+pV. The enthalpy change, ∆H, for finite changes at constant pressure is given by the expression ∆H=∆U+p∆V, so making the enthalpy change for a process equal to the heat exchange in a system at constant pressure. For a chemical system which releases or absorbs a gas at constant pressure, the enthalpy change is related to the internal energy change by ∆H=∆U+∆n.RT, where ∆n is the molar change in gaseous component.
Enthalpy is a state function whose absolute value cannot be known. ∆H can be ascertained, either by direct methods, where feasible, or indirectly. An increase in the enthalpy of a system, for which ∆H is positive, is referred to as an endothermic process. Conversely, loss of heat from a system, for which ∆H has a negative value, is referred to as an exothermic process. The enthalpy change arising from a temperature change at constant pressure is given by the expression ∆H=Cp∆T, providing that Cp
does not appreciably change over the temperature range of interest. Where Cp does change, the integral form of the equation,
, is used. In a chemical reaction, the enthalpy change is equal to the difference in enthalpy between the reactants and products:
∆HReaction=ΣH(Products)−ΣH(Reactants). The value of ∆H for a reaction varies considerably with temperature. Kirchhoff's s equation, derived from the properties of enthalpy, quantifies this variation. Where Cp does not appreciably change over the temperature range of interest, it may be expressed in the form ∆HT2−∆HT1=∆Cp∆T, or as
where ∆Cp is a function of temperature.
The first law (B1) Entropy and change (B5) Thermochemistry (B3) Free energy (B6) Related topics
Entropy (B4) Statistical thermodynamics (G8)
Enthalpy
The majority of chemical reactions, and almost all biochemical processes in υiυo, are performed under constant pressure conditions and involve small volume changes. When a process takes place under constant pressure, and assuming that no work other than pV work is involved, then the relationship between the heat changes and the internal energy of the system is given by:
dU=dq−pexdV (infinitesimal change) ∆U=q−pex∆V (finite change) The enthalpy, H, is defined by the expression; H=U+pV, Hence for a finite change at constant pressure:
∆H=∆U+ pex∆V
Thus, when the only work done by the system is pV work,
∆H=q at constant pressure
Expressed in words, the heat exchanged by a system at constant pressure is equal to the sum of the internal energy change of that system and the work done by the system in expanding against the constant external pressure. The enthalpy change is the heat exchanged by the system under conditions of constant pressure.
For a reaction involving a perfect gas, in which heat is generated or taken up, ∆H is related to ∆U by:
∆H=∆U+∆n RT
where ∆n is the change in the number of moles of gaseous components in the reaction.
Hence for the reaction , (1
mole of gaseous CO2 is created), and so ∆H=∆U+2.48 kJ mol−1 at 298 K.
Properties of enthalpy
The internal energy, pressure and volume are all state functions (see Topic B1), and since enthalpy is a function of these parameters, it too is a state function. As with the internal energy, a system possesses a defined value of enthalpy for any particular system at any specific conditions of temperature and pressure. The absolute value of enthalpy of a system cannot be known, but changes in enthalpy can be measured. Enthalpy changes may result from either physical processes (e.g. heat loss to a colder body) or chemical processes (e.g. heat produced υia a chemical reaction).
An increase in the enthalpy of a system leads to an increase in its temperature (and υice υersa), and is referred to as an endothermic process. Loss of heat from a system lowers its temperature and is referred to as an exothermic process. The sign of ∆H indicates whether heat is lost or gained. For an exothermic process, where heat is lost from the system, ∆H has a negative value. Conversely, for an endothermic process in which heat is gained by the system, ∆H is positive. This is summarized in Table 1. The
sign of ∆H indicates the direction of heat flow and should always be explicitly stated, e.g.
∆H=+2.4 kJ mol−1.