Ch 1: Definitions (continued). Homogeneity If every intensive variable has the same value for every point, the system is homogenous
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Equilibrium State can be:
a) Stable equilibrium: When small variations about equilibrium do not take system away from that state
b) Unstable equilibrium:When small variations about equilibrium change the state
c) Metastable: System is stable w.r.t. small variations of some properties, but unstable w.r.t. small variations in other properties
1st Example: supercooled water in mechanical and thermal equilibrium with environment. If we introduce a small piece of ice to the system, the whole system will spontaneously freeze
2nd Example: mixture of H2 and O2 at room temperature. A spark will cause an explosive chemical reaction throughout the system leading to the formation of water
Two example systems were in equilibrium w.r.t. freezing or chemical reactions until introduction of ice or spark
1. Reversible – when successive states between initial and final states along the path differ by an infinitesimal amount from equilibrium
2. Irreversible – if the change from final back to initial cannot be done without changing the environment
Ch 2: Exact Differentials
Given a function z=f(x,y), its exact differential can be written like:
Let z be a differential expression of type
where x and y are independent variables, and M and N are coefficients and functions of x,y. If we integrate (1), we will have
which are meaningless unless a functional relation f(x,y) = 0 is known. This relation specifies a path in the x-y plane along which the integration must be performed. This is called a line integral and the result depends on the path.
If we have that
in which case
and then z is called an exact or total differential. If this is the case, the integration of (4) will yield
where C is a constant and z is a point functionthat depends only on the pair of values (x,y) and C
When the integral of (5) is taken along a closed path
In order to check if eq. (1) obeys condition (3), we would need to know the function z. It is easier to apply the theorem of crossed derivatives, assuming that z and its 1st derivatives are continuous.
Eq. (8) is a necessary condition for eq. (3) to hold. It is also sufficient for the existence of a function that obeys it, because we can always find a function
Eq. (8) satisfies eq. (3). Equations (3), (5), (6), (7), and (8) are equivalent conditions that define z as a point function.
If z is not an exact differential, an integrating factor, , can be found such that z = du is an exact differential.
Importance: state functions, like internal energy, are point functions of state variables. The three primary state variables (dp, dV, dT) are exact differentials.
Example 1: Let z = 2ydx + xdy that we want to integrate between x = 0, y = 0 and x = 2, y = 2 and we choose 2 arbitrary paths (a) and (b).
(a) The path is defined by y = x, then z = 3xdx. This can be integrated between the two limits giving
(b) Increasing x from 0 to 2 at y = 0. The integration of z along this path is 0. Then increase y to 2 while keeping x = 2. This will give 4, which is the total change between the two points, different from (a)!
Obviously, no point function can be defined from δz.