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Ch 1: Definitions (continued)

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- Homogeneity
- If every intensive variable has the same value for every point, the system is homogenous
- If there are several portions (or sets of portions) each of them homogeneous, but different from each other, each homogeneous portion (or set) is called a phase and the system is called heterogeneous (vapor, liquid, ice)
- If the values of intensive properties change in a continuous manner from point to point, the system is said to be inhomogeneous

- Equilibrium State
- Condition – Environment must remain unchanged
- Equilibrium exists when system properties do not change with time (stationary state)
Equilibrium State can be:

a) Stable equilibrium: When small variations about equilibrium do not take system away from that state

- Equilibrium State
b) Unstable equilibrium:When small variations about equilibrium change the state

- Example: small cloud droplet in “equilibrium” with surrounding water vapor at constant pressure
- Slight variation in vapor pressure around droplet will cause the droplet to grow by condensation or shrink by evaporation with the process continuing (we will examine this later)

- Equilibrium 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

- Equilibrium State
- Stable equilibrium asserts that if means are found to cause small variations in the system, this will not lead to a general change in its properties
- Lack of equilibrium leads to mechanical changes, chemical reactions or changes of physical state, or changes in thermal state
- So we speak of mechanical, chemical, or thermal equilibrium

- Equilibrium State
- Mechanical equilibrium – the force exerted by the system is uniform throughout the system and is completely balanced by external forces
- Chemical equilibrium – the internal structure and the chemical composition of the system remain unchanged
- Thermal equilibrium – the temperature is uniform throughout the system and is the same as that of the surroundings

- Transformation: A transformation takes a system from an initial state (i) to a final state (f) along some path
1. Reversible – when successive states between initial and final states along the path differ by an infinitesimal amount from equilibrium

- Can be reversed anywhere along the path so that the system & its environment return to their initial states
- In practice, external conditions change slowly, allowing system enough time for adjustment to new conditions

- Transformation
2. Irreversible – if the change from final back to initial cannot be done without changing the environment

- Cyclic transformation goes:
- Types of transformations
- Isothermal– path is an isotherm (T = const.)
- Isobaric – path is constant pressure line (surface)
- Isochoric – path is of constant volume line
- Adiabatic – the transformation does not involve the system exchanging heat with its environment (is not isothermal)

- THERMODYNAMICS
- The study of equilibriumstatesof a system that has been subjected to some energytransformation
- Basically – the consideration of transformations of heat into mechanical work and mechanical work into heat

- System of Units
- Based on choice associated with 3 fundamental quantities – length, mass, and time
- MKS system – Meter, Kilogram, Second
- Meter – 1,579,800.298728 wavelengths of light from a He-Ne laser (actually distance light travels in a vacuum during a time interval of 1⁄299,792,458 of a second)
- Kilogram – mass of standard body of Pt (90%)-Ir (10%) alloy kept at Int’l. Bureau of Weights & Measures, Paris
- Second – duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom (at 0K)

- SI Units – MKS and kelvin [K] for thermodynamic temperature

- System of Units
- Other units commonly used, but not recommended
- Bar (bar): 1 bar = 105Pascals (Pa)
- Millibar (mb): 1 mb = 102 Pa
- mm Hg (torr): 1 torr = 133.322 Pa
- Atmosphere (atm): 1 atm = 1.01325 × 105 Pa = 760 torr
- Pound/sq. inch (p.s.i): 1 p.s.i. = 6894.76 Pa

- Convenient pressure unit (equivalent to mb) would be hPa
- Energy – Int’l. Steam Table calorie (IT cal) = 4.1868 J
- SI units should be used whenever possible and in all calculations, where units should always be included

- Other units commonly used, but not recommended

Ch 2: Exact Differentials

- I) Path not important for i f:
- Thermodynamic property is a point function
- Thermodynamic property is an exact differential

- Thermodynamic property is not a point function
- Thermodynamic property is not an exact differential

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

or

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

exact differential:

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.

equivalent condition:

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.