The state and state functions

Contents

• The microscopic state
• The equilibrium state
• The macroscopic description of an equilibrium state
• The simple system
  • Equations of state
  
  
• State functions
• Quantities which are not state functions
• Extensive and intensive variables
• About this document ...

The microscopic state

We would like to be able to describe the state of our system, i.e. to characterize the system completely, so that we know everything there is to know about it. One way of doing this is to specify the microscopic state by giving the positions and velocities of all the atoms in the system. If the system contains N atoms, this microscopic description would involve giving 6N numbers, three describing the position and three the velocity of each atom. Even our largest supercomputers would have trouble storing this amount of data for N as small as one trillion. Thus, for macroscopic systems we need an alternative description.

The equilibrium state

A special kind of state of a system is the equilibrium state, which is attained when all system properties exhibit stable values that do not change with time. A simple way of reaching equilibrium is to isolate our system from the surroundings and leave it alone. Initially, the temperature, pressure, density etc. may vary with time. After a sufficiently long time has elapsed the system will reach equilibrium. It is easy to prove that in a system at equilibrium the temperature, pressure and concentrations of each species must be the same throughout the sample.

The macroscopic description of an equilibrium state

The truly amazing observation is that a macroscopic system at equilibrium may be fully characterized by specifying values of only a few variables, such as the amount of substance (n), mass (m), pressure (p), temperature (T) or volume (V). This specifies the macroscopic state of the system.

For example, in order to estimate how far we can drive our car before re-filling the gas tank, we only need to know the current volume of gasoline in our tank. We do not need to know the positions of all the molecules in the tank, the tank's shape or the life story of the gas station attendant.

An example of a system not at equilibrium is a cup of coffee to which we have just added sugar and cream. Until we mix things thoroughly, it is not enough to know how much coffee, cream and sugar there is in our cup to describe the state. We will have swirls of cream, sugar distributed mostly at the bottom, etc. Another example of a system not an equilibrium is a human body. Life would not be possible if the concentrations of all biomolecules were the same throughout our bodies!

The simple system

For the simplest kind of system, known as the one-component or simple system, such as a cup of pure liquid water, we need only three variables to specify the state (e.g. n, p and T).

For more complicated systems, such as mixtures, we need to specify p, T and the amounts of each component (e.g. the amounts of water and glucose in an aqueous glucose solution).

Equations of state

We can come up with more than three variables to describe a one-component system, e.g. n, p, T, V. Since the three variables (n,p,T) are sufficient to describe the state (i.e. three variables are independent), it must be possible to calculate any other system properties from the basic three. This leads to the conclusion that there must exist relations between the different system properties. These relations are called equations of state. For example, since specifying (n,p,T) completely defines the state of a simple system, this state must have a definite volume. We write this in the form of an equation:

$$V =  V(n,p,T)$$

Of course, alternative ways are also possible. E.g. if we know the values of (n,p,V) of the system, it's state is also fully characterized and it must have a definite temperature:

$$T =  T(n,p,V)$$

and if we know the values of (n,V,T) then p has a definite value

$$p =  p(n,V,T)$$

For real substances the relations between T, p and V have to be determined experimentally, by performing a series of measurements. There is a hypothetical substance which has an especially simple equation of state - the ideal gas, describe below.

Examples.
Examples of equations of state are the ideal gas equation

$$pV =  nRT$$

the van der Waals equation for real gases

$$p =  \frac{nRT}{V - nb} -  a \left(\frac{n}{V}\right)^2$$

and this equation true for solids and liquids

$$V \approx  const    (liquids and solids)$$

when small changes of T and p are considred.

State functions

A state function is any quantity X which has a definite value for each state of the system

$$state k \rightarrow  X_k$$

Some simple quantities, such as n, p, V, T are evidently state functions. All this means is that if we know the state of our system, then the values of these quantities are determined:

$$state k \rightarrow  n_k, p_k, V_k, T_k$$

In the case of the internal energy U we used microscopic arguments to convince ourselves that U is also a state function. See the Energy section.

For any state function X we can define the change of X in a process

$$X_i    (the value of X in the initial state i)$$

$$X_f    (the value of X in the final state f)$$

$$\Delta X =  X_i -  X_f     (the change of X in the process)$$

For the special case of a cyclic process, where the initial and final states are the same

$$\Delta X =  X_i -  X_i =  0$$

Quantities which are not state functions

A number of quantities are not state functions. Heat and work depend on the detailed path of the process, not just the end points. The magnetization of a ferromagnetic sample depends not only on the current value of the magnetic field but also on the history of the sample (magnetic hysteresis).

This simple example illustrates the concept of a state function. Consider a mountain with a parking lot at the bottom and a lodge at the top. There are two trails from bottom to top. The first trail, short and difficult, goes straight up the mountain. The second trail, longer and less steep, spirals around the mountain. The distance traveled and the sights you see will be different on the two trails. However, one quantity, the difference in elevation between the top and bottom, is the same for both trails, as it depends only on the starting and ending point of the trails.

Extensive and intensive variables

Extensive variables are proportional to the amount of substance. These are such quantities as the amount of substance n, the mass m, volume V, internal energy U, heat capacity C, enthalpy H, entropy S or free energy G.

Intensive variables have values independent of the amount of substance. These are quantities such as pressure p, temperature T or density d.

To understand the meaning of the terms extensive and intensive, imagine that we prepared a container of liquid water at pressure p and temperature T, and divided it in half with some sort of separator. In each of the halves the temperature would still be T and the pressure p. However, the amount of substance, mass, volume, internal energy of each half would be one half that of the whole.

Here is how to make a new intensive quantity : divide an extensive quantity by n or m. Thus we get the density

$$d =  \frac{m}{V}$$

molar volume

$$V_m =  \frac{V}{n}$$

molar heat capacity (at p=const)

$$C_{P,m} =  \frac{C_P}{n}$$

molar internal energy

$$U_m =  \frac{U}{n}$$

Here is how to get a new extensive quantity : multiply an extensive quantity by an intensive quantity. Thus the products pV and TS are extensive, and quantities such as

$$H =  U +  pV$$

$$G =  H -  TS =  U +  pV -  TS$$

are also extensive.

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