PROPERTY DIAGRAMS AND STEAM TABLESThermodynamicsTemperature-Entropy (T-s) DiagramA  T-s  diagram  is  the  type  of  diagram  most  frequently  used  to  analyze  energy  transfer  systemcycles.   This is because the work done by or on the system and the heat added to or removedfrom the system can be visualized on the T-s diagram.   By the definition of entropy, the heattransferred to or from a system equals the area under the T-s curve of the process.   Figure 13 isthe T-s diagram for pure water.   A T-s diagram can be constructed for any pure substance.   Itexhibits the same features as P-u  diagrams.Figure 13   T-s Diagram for WaterIn the liquid-vapor region in Figure 13, water and steam exist together.   For example, at pointA, water with an entropy (s_f) given by point B, exists together with steam with an entropy (s_g)given by point C.  The quality of the mixture at any point in the liquid-vapor region can be found by .s = xs_g  + (1 - x)s

Amagat's law or the Law of Partial Volumes of 1880 describes the behaviour and properties of mixtures of ideal (as well as some cases of non-ideal) gases. Of use in chemistry and thermodynamics, Amagat's law states that the volume V_m of a gas mixture is equal to the sum of volumes V_i of the K component gases, if the temperature T and the pressure p remain the same: ^[1]

 .

In chemistry and physics, Dalton's law (also called Dalton's law of partial pressures) states that the total pressure exerted by a gaseousmixture is equal to the sum of the partial pressures of each individual component in a gas mixture. This empirical law was observed by John Dalton in 1801 and is related to the ideal gas laws.

Mathematically, the pressure of a mixture of gases can be defined as the summation

       or      

where  represent the partial pressure of each component.

It is assumed that the gases do not react with each other.

where  the mole fraction of the i-th component in the total mixture of m components .

When a real gas, as differentiated from an ideal gas, expands at constant enthalpy (i.e., no heat is transfered to or from the gas, and no external work is extracted), the gas will be either cooled or heated by the expansion. That change in gas temperature with the change in pressure is called the Joule-Thomson coefficient and is denoted by µ, defined as:

µ = (dT/dP) at constant enthalpy

The value of u depends on the specific gas, as well as the temperature and pressure of the gas before expansion. For all real gases, µ will equal zero at some point called the "inversion point". If the gas temperature is below its inversion point temperature, µ is positive ... and if the gas temperature is above its inversion point temperature, µ is negative. Also, dP is always negative when a gas expands. Thus:

If the gas temperature is below its inversion temperature:

-- µ is positive and dP is always negative

-- hence, the gas cools since dT must be negative

If the gas temperature is above its inversion temperature:

-- µ is negative and dP is always negative

-- hence, the gas heats since dT must be positive

"Perry's Chemical Engineers' Handbook" provides tabulations of µ versus temperature and pressure for a number of gases, as do many other reference books. For most gases at atmospheric pressure, the inversion temperature is fairly high (above room temperature), and so most gases at those temperature and pressure conditions are cooled by isenthalpic expansion.

Helium and hydrogen are two gases whose Joule-Thomson inversion temperatures at atmospheric pressure are very low (e.g., about −222 °C for helium). Thus, helium and hydrogen will warm when expanded at constant enthalpy at atmospheric pressure and typical room temperatures.

It should be noted that µ is always equal to zero for ideal gases (i.e., they will neither heat nor cool upon being expanded at constant enthalpy).

the definition of entropy is given by

      

Rearranging the above equation gives

                      (1)

The entropy change during an internally reversible process (1-2) is

      

Only when the relation between δQ and T is known, the entropy change can be determined. The relations between δQ and T can be found by considering the energy balance of a closed system.

The differential form of the energy balance for a closed system, which contains a simple substance and undergoes an internally reversible process, is given by

      dU = δQ_rev - δW_rev             (2)

The boundary work of a closed system is

      δW_rev = PdV                     (3)

Substituting equations (1) and (3) into equation (2) gives

      dU = TdS- PdV
      TdS = dU + PdV

or

      Tds = du +Pdv                   (4)

where
      s = entropy per unit mass

Equation (4) is known as the first relation of Tds, or Gibbs equation.

The definition of enthalpy gives       h = u + Pv differential the above equation yields       dh = du +Pdv + vdP Replacing du + Pdv with Tds yields       dh = Tds + vdP       Tds = dh -vdP                   (5) Equation (5) is known as the second relation of Tds. Although the Tds equations are obtained through an internally reversible process, the results can be used for both reversible or irreversible processes since entropy is a property. Rewriting equations (4) and (5) in the following form       ds = du/T + Pdv/T       ds = dh/T + vdP/T The entropy change during a process can be determined by integrating the above equations between the initial and the final states.

The vaporization curves of most liquids have similar shape. The vapour pressure steadily increase as the temperature increases. A good approach is to find a mathematical model for the pressure increase as a function of temperature. Experiments showed that the pressure P, enthalpy of vaporization, DH_vap, and temperature T are related,P = A exp (- DH_vap / R T)where R (= 8.3145 J mol^-1 K^-1) and A are the gas constant and unknown constant. This is known as the Clausius- Clapeyron equation. If P₁ and P₂ are the pressures at two temperatures T₁ and T₂, the equation has the form:

     P1    DHvap    1     1 ln (---) = ----  (--- - ---)      P2     R      T2    T1 

The Clausius-Clapeyron equation allows us to estimate the vapor pressure at another temperature, if the vapor pressure is known at some temperature, and if the enthalpy of vaporization is known.

As we have seen, the fundamental thermodynamic relation  implies that the natural variable in which to express  are  and : .

That means that on purely mathematical grounds, we can write 

But comparison with the fundamental thermodynamic relation, which contains the physics, we can make the following identifications: 

These (especially the second) are interesting in their own right. But we can go further, by differentiating both sides of the first equation by  and of the second by : 

Using the fact that the order of differentiation in the second derivation doesn't matter, we see that the right hand sides are equal, and thus so are the left hand sides, giving 

By starting with ,  and , we can get three more relations.

The two equations involving derivatives of  are particularly useful, as they provide a handle on  which isn't easily experimentally accessible.

For non-hydrodynamic systems, we can obtain analogous relations involving, say,  and  instead of  and ; for instance by starting with we get .