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(1)We have agreed that work, potential energy, kinetic energy, and heat are all forms of energy. Historically, it was not obvious that heat belonged in this list. But beginning with the experiments of Count Rumford of the Holy Roman Empire, and later the experiments of Joule, it became clear that heat, too, was just another form (or manifestation) of energy.
Recall that we defined the internal energy, U, as the total energy of the system. (Although the existence of atoms and molecules is not relevant to thermodynamics, we said that the internal energy is the sum of all the kinetic and potential energies of all the particles in the system. This statement is outside the realm of thermodynamics, but it is useful for us to gain an intuitive "feel" for what the internal energy is.)
Recall also that energies are always measured relative to some origin of energy. The origin is irrelevent to thermodynamics because we will always calculate changes in U and not absolute values of U. That is, we calculate
In words, this equation reads, "the change in the internal energy is equal to the final internal energy minus the initial internal energy." This equation also reminds us that U is a "state function." That is, the change in U does not depend on how the change was done (in other words, on the path), but depends only on the initial and final states.
Recall that the first law of thermodynamics in equation form for a finite change, is given by,
Equation (2) tells something else of importance. We know that U is a state function and that ΔU is independent of path. However, w is not a state function so that w depends on path. Yet the sum of w and q is path independent. The only way this can happen is if q is also path dependent. We now see that we are dealing with two path-dependent quantities, q and w.. (2)
For a differential change we write the first law in differential form,
The w in Equation (2) or the dw in Equation (3)3 includes all types of work, work done in expansion and contraction, electrical work, work done in creating new surface area, and so on. Much of the work that we deal with in thermodynamics will be work done in expansion and contraction of the system, or pV work. Recall that the expression for pV work is,. (3)
If we want to include both pV work and other types of work we can write the first law as,. (4)
Let's now confine ourselves to systems where there is only pV work. In this case the first law can be written,. (5)
Suppose we now regard U as a function of T and V. That is, U = U(T,V). Then, for dU we can write,. (6)
For a process at constant V (dV = 0) Equations (6) and (7) become,. (7)
and(8)
We know, from our discussion on heat and heat capacity , that the differential heat at constant volume can also be written as,. (9)
so,(10)
Comparing Equations (9) and (11), and recognizing that the change dUV is the same in both cases, we see that,. (11)
We shall regard Equation (12) as the formal thermodynamic definition of the heat capacity at constant volume. This new definition is more satisfactory than our previous temporary definition,. (12)
Equation (12) is a better definition of the heat capacity because it is usually more satisfactory to define thermodynamics quantities in terms of state functions, like U, T, V, p, and so on, rather than on things like q and w which depend on path.. (13)
One other comment, we can integrate Equation (8), at constant volume, to get,
In words, for any process at constant volume the heat, q, is the same as the change in the internal energy, ΔU.. (14)
Enthalpy
It turns out that V is not the most convenient variable to work with or to hold constant. It is much easier to control the pressure, p, on a system than it is to control the volume of the system, especially if the system is a solid or a liquid. What we need is a new function, with units of energy, which contains all the information that is contained in U but which can be controlled by controlling the pressure. Such a function can be defined (created) by a Legendre transformation. There are particular criteria which must be met in making a Legendre transformation, but in our case here these criteria are met. (A full discussion of the mathematical properties of Legendre transformations is beyond the scope of this discussion. There are more details given in the Appendices to Alberty and Silby.) In our case we will define a new quantity, H, called the enthalpy, which has units of energy, as follows,
We can show that H is a natural function of p (in the same sense that U is a natural function of V) as follows,. (15)
One of the great utilities of the enthalpy is that it allows us to use a state function, H, to describe the heat involved in processes at constant pressure rather than the heat, q, which is not a state function. To see this, let's go through the same process with dH that we did with dU above. Let's regard H as a function of T and p (for now). Then we can write,. (16a, b, c)
Consider a process at constant pressure (dp = 0). From Equation (16c) we conclude that.. (17)
and from Equation (17) we get,(18)
We know, from our discussion on heat and heat capacity , that the differential heat at constant pressure can also be written as,. (19)
so,(20)
Comparing Equations (19) and (21), and recognizing that the change dHp is the same in both cases, we see that,. (21)
We shall regard Equation (22) as the formal thermodynamic definition of the heat capacity at constant pressure. Again, this definition is much more satisfactory than our previous temporary definition,. (22)
since it defines the heat capacity in terms of the state function, H, rather than in terms of q which is not a state function., (23)
Just as we integrated equation (8), we can integrate Equation (21), at constant pressure, to get,
In words, for any process at constant pressure the heat, q, is the same as the change in enthalpy, ΔH. This equation contains no approximations. It is valid for all process at constant pressure. Equation (24) is vastly more useful than its counterpart at constant volume because we carry out our chemistry at constant pressure much more often than we do at constant volume.. (24)
People sometimes ask, "What is the meaning of H?" Unfortunately, there is no simple, intuitive physical description of enthalpy like there is for the internal energy. (Recall that the internal energy is the sum of all kinetic and potential energies of all the particles in the system). The nearest thing we can come to as a description of H is the one above where ΔH is the heat (gain or loss) in a constant pressure process. For this reason the enthalpy is ocassionally referred to as the "heat content."
Reminder: Nuclear energy was unknown to the original formulators of
thermodynamics. We now know that matter can be converted into energy and
vice versa. The "energy equivalent of matter" is given by the famous Einstein
formula, E = mc2, where m
is the mass of the matter and c is the velocity of light. Since
the velocity of light is very large, about 3 x 108
m/s, a small amount of mass is equivalent to a very large amount of energy.
Strictly speaking, the statement, "energy is conserved," should be replaced
by the statement, "energy plus the energy equivalent of mass is conserved."
That is, energy + mc2 is conserved.
The conversion of mass to energy or energy to mass in chemical reactions
is so small that it is virtually never observed in chemical problems. So,
for chemical thermodynamics, the simpler statement that energy is conserved
is sufficient.
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