Exact Differetial Theorem

To give a clear representation of a state function, it is often helpful to investigate the two theorems on exact differentials.

Thermodynamics Definitions

System

A system is any region that occupies a volume and has a boundary. The volume outside of this boundary is referred to as the surroundings of the system. The sum of the system and its surroundings is called the universe. Thermodynamics considers systems only at the macroscopic level. Systems can be divided into three general types:

• Isolated System: A system that does not have any mass and energy exchange with the surroundings. An example includes the universe.
• Closed System: This is a system where only energy in the form of heat and work is exchanged with the surroundings. No mass is exchanged. 
• Open System: This is a system where both mass and energy is exchanged with the surroundings. 

Since the equations available to analyze closed and open systems differ from one another, it is important to properly define a system.

State

In order to describe a system we need to know the quantities that characterize it. These quantities are called properties and include volume, mass, temperature, pressure, etc. 

A complete list of the properties of a system describe its state.

Intensive and Extensive Properties

Thermodynamic properties are considered to be either intensive or extensive. A property is considered to be extensive if it is proportional to the mass of the system e.g., volume, kinetic and potential energy. An intensive property is independent of the systems mass e.g., viscosity, density, temperature, pressure, mole fraction and refractive index.

An easy way to visualize this is to divide a system into two equal parts. Each of the part will have the same value of intensive properties (T , P , ρ) as the original system, but will have half the value of the extensive property (V) as shown in the figure below. In short, extensive properties are additive while intensive properties are not.

Specific properties are extensive properties divided by the total mass or total moles of the system, i.e.,

Note: All specific properties are intensive.

The Degrees of Freedom are the number of independent intensive variables needed to specify the state of the system. The Gibbs phase rule specifies the number of degrees of freedom (F) for a system at equilibrium and is expressed as follows:

P + F = C + 2

where P is the phase number and C is the number of components.

Thus, is you are working with a single phase, one component system, you need to specify two independent intensive properties. Two properties are independent if one property can be varied while the other is constant. For example, temperature and density are always independent properties and together they can fix the state of a single-phase, single-component system.

Equations of State

This can be defined as any mathematical relationship between the variable T, P and V.

 

Process

Process is a change of state that can occur in numerous ways. Work and heat can only occur during processes and only across the boundary of the system. The curve describing the process is called a process path. If the final state is the same as initial state, then the overall process is called a cyclic process.

State and Path Functions

State functions are an important thermodynamic concept. If the magnitude of a thermodynamic property depends only on the initial and final states and is independent of the path being followed then its known as a state function. Otherwise, it’s called a path function. All the thermodynamic variables are state functions except heat and work.

Some characteristics of state functions are:

• State functions can be solved using integral and differential calculus.
• If a state function φ, undergoes a cyclic process, its initial and final values are the same (Δφ = 0), and this will be true regardless of the path being used to carry out the cyclic process.

Some characteristics of path functions are:

• Cannot be solved using integral and differential calculus
• If a path function φ, undergoes a cyclic process, its initial and final values:

(i). will be different (Δφ ≠ 0),

(ii). will depend on the cyclic path being used,

(iii). will be different for every path.

Steady-state

This means the dependent variable does not change as a function of time. If the dependent variable is φ, then:

Uniform

This means the dependent variable is not a function of position. This means that all three of the partial derivatives with respect to position be zero, i.e.,

The variation of a physical quantity with respect to its position is called a gradient. Thus, the gradient of a quantity must be zero for a uniform condition to exist.

Equilibrium

A system is considered to be at equilibrium if both steady-state and uniform conditions are met simultaneously. This means that the systems properties like temperature, density, pressure are constant at all time. Equilibrium has the following characteristics:

·       No work is done by a system in equilibrium.

·       Its state is completely specified when a given number of independent state functions are specified.

Equilibrium can be classified into 4 classes that can easily be conceptualized using the analogy of a ball on a solid surface being acted on by gravity.

In thermodynamics only systems with stable equilibrium states are considered in their initial and final stable equilibrium states to determine the heat and work interactions with its surroundings.

 

 

 

Process Calculations

Describing Physical Quantities

Units

the metric and America engineering systems are the most widely used unit systems in the industrial world. However, a single worldwide unit system was introduced in order to solve any problems that may arise when products are traded internationally from one unit system to another. Hence the Systeme Internatinale d’Units or SI system was formed.

Some widely used systems are summarized below

System	Mass	Length	Time	Temperature
SI	kg	m	s	Kelvin
American	lbm	ft	s	Fahrenheit
cgs	g	cm	s	Celcius

 

Conversion Factors

This is a relationship expressed by an equation where the entries on each side of the equation are the same quantity but expressed in a different form of units.

for example: Convert 3 meters to inches.

Moles

This is the number of molecules in a compound whose mass in grams is numerically equal to its molecular weight. can be in the form of gram-mol (gmol), kilogram-mol (kgmol) or pound-mole (lbmol) depending one your preferred unit system.

Mixture Composition

it is important to find the amount of a particular substance in a mixture using concentration (c) of the substance. The following relationships can help find the concentration, mass and mole fractions of a substance, A.

The above equations ca be manipulated to suit the problem being solved. Depending on what variables are provided, it is easy to solve for unknowns as follows (where MW stands for molecular weight of a species):

Dimensional Consistency

1.    Terms that are added or subtracted together MUST have the same units. For example, in the equation y = mx + b, the units of mx and b must be the same.

2.    Exponents MUST be unitless, therefore an exponent with multiple terms must have those terms cancel out.

Equations that correctly describe a physical phenomenon MUST obey the above rules of dimensional consistency

 

Exercise

1.    Convert the following

a)    an acceleration of 1 cm/s² to km/yr² (Ans: 9.95 * 10⁹ km/yr²)

b)    1 bar into g/cm-s² (Ans: 10⁶ g/cm-s²)

c)    10,000 g/cm²-s² into J (Ans: 0.001 J)

2.    Calculate the Pressure at the bottom of a swimming pool with a depth of 2 meters.

 

3.    When a fluid flows from one location to another under certain circumstances, the changes in fluid properties can be described by Bernoulli equations:

where α: dimensionless correction factor, ρ: fluid density, P: fluid pressure, v: fluid velocity, z: fluid elevation

Show that this equation is dimensionally consistent in the SI system.

4.    A gas mixture has the following mass percentage: 70% N₂, 14% O₂, 4% CO and 12% CO₂. What are the mole percentages of the gases in the mixture? what is the average molecular weight of the mixture? (MW for N₂ = 28g/mol, O₂ = 32 g/mol, CO = 28 g/mol and CO₂ = 44 g/mol).

Assume a mass basis of 100 g for the total mixture

components	mass fraction	mass (g)	MW (g/mol)	Moles	Mole fraction	Mole Percent
N2	0.70	70	28	2.500	0.745	74.5
O2	0.14	14	32	0.432	0.131	13.1
CO	0.04	4	28	0.143	0.043	4.3
CO2	0.12	12	44	0.273	0.081	8.1
Total	1.00	100		3.354	1.00	100

 

Average molecular weight