**Avogadro's Law **

A mole of a substance has a mass numerically equal to the molecular weight of the substance.

One g mol of oxygen has a mass of 32gm, 1kg mol of oxygen has a mass of 32kg, 1kg mol of nitrogen has a mass of 28kg, and so on.

Avoadro's law states that the volume of a g mol of all gases at the pressure of 760mm Hg and temperature of 0 deg C is the same, and is equal to 22.4 litres. Therefore, 1g mol of a gas has a volume of 22.4x10^{3} cm^{3} and 1kg mol of a gas has a volume of 22.4m^{3} at a normal temperature and pressure (NTP).

For a certain gas, if m is its mass in kg, and its molecular weight, then the number of kg moles of the gas, n would be given by:

The molar volume, v, is given by:

Where V is the total volume of the gas in m3.

(Moles = wt/mol.wt)

For more information on the specific heats, internal energy and enthalpy of gases, click here and scroll to the section at the bottom.

**Real Gases**

Gases which do not follow closely the ideal gas equation are known as real gases. In order to find the relationship between different properties, the obvious approach is to attempt to produce a state equation, such as the Van der Waals equation. But these equations require time consuming calculations in order to evaluate fluid properties. Another approach which is simpler but less accurate is to use the fact that the shape of P - V - T diagram is similar for most pure substances, if you scale their properties by the critical point values.

This is known as the **law of corresponding states**.

**The Principle of Corresponding States**

**If we reduce all P - V - T surfaces to a common scale based on their critical properties, then sealed points on these surfaces should be in corresponding states.**

So we should be able to map properties from a known P - V - T surface onto an unknown one. This idea works well in the vapour - gas region but does not adequately represent the two phase and liquid areas on the P- V - T surfaces.

** **

**Reduced Properties**

The primary properties are expressed as ratios of their critical values i.e.

*Examples**:*

Calculate by the law of corresponding states the specific volume of ammonia at 50 ^{o}C and 20 bar (dry-saturated).

*The critical states are:*

NH3: Pc = 113 bar, tc = 132 ^{o}C, vc = 0.00429 m3/kg

H2O: Pc = 221.2 bar, tc = 374 ^{o}C, vc = 0.00317 m3/kg

*The reduced properties are:*

_For similar reduced properties of steam:

P = 0.177 ´ 221 = 39.1 bar

T = 0.8 ´ 647 = 517 K or 244 ^{o}C

and v at this condition is approximately on the saturation line at 0.05 m3/kg

For ammonia:

v = vR vC = 15.6 0.00429 = 0.067 m3/kg

The actual value from ammonia tables is 0.0635 m3/kg .

An error of 5.5%

**Compressibility Factors & Charts**

The perfect gas equation of state may be extended to include non-ideal gases by introducing a compressibility factor Z thus:

Pv = ZRT

The compressibility factor is a function of the pressure and depends on the gas. For any state defined by the pressure and temperature the specific volume may be accurately found if the appropriate values of Z are known. Individual compressibility charts would then be needed for each fluid based on experimental data. An example for Nitorgen is shown below:

However, if the compressibility factor approach is used in conjunction with the law of Corresponding States we can plot the **Generalised Compressibility Chart**.

**Comments On The Generalised Compressibility Charts**

(i ) The charts show that a very low pressure Z1, and the gas behaves like an ideal gas.

(ii) The ideal gas equations are relatively accurate when the temperature is equal to or greater than twice TR. However, this does not seem to work above PR of about 5 or 6.

(iii) The chart indicates that the value of Z at the critical point Zc is 0.2 for all gases. But in fact it ranges between 0.19 to 0.35.

(iv) A pressure of 1 bar for most of the permanent gases corresponds to a reduced pressure of 0.1 or less and the ideal gas relations will hold with acceptable accuracy for all normal temp range.

(v) the deviation from ideal gas equation is greatest near the critical point.

** **

**Accuracy**

Published literature indicate that the charts are accurate to within ± 1% for most gases and ± 5% if hydrogen, helium, neon, ammonia and water are included.

Of course if individual compressibility charts or tables are available these should be used instead.

The perfect and ideal gases

The development of the equation of state for a perfect gas, PV = MRT, comes naturally from two sources. Firstly the experimental evidence given by Boyles and Charles law and secondly from the kinetic theory of gases. We are very fortunate that we are in a world where some conditions of engineering interest can be represented by such a simple equation. In general ideal gas can be divided into perfect and semi-perfect gases. In order to strictly define a perfect gas we need to introduce two new proprties of gases. These are the specific heats at constant pressure and constant volume.

**Specific Heats Of Gases**

Now we know that specific heat is the amount of heat required to raise the temperature of 1kg of a substance by one degree,i.e

Now from First Law of Thermodynamics:

Assuming work done is displacement work only and that process is quasi-static then:

dQ=dU+PdV

or in terms of specific quantities (i.e. per kg): dq=du+pdv

For constant volume process:

Similarly for a constant pressure process we know that dq = dh and so:

Since the expression for Cp and Cu contain properties we can conclude that Cp and Cv themselves **are properties**.

We can measure Cp and Cu at any particular state by drawing on an h - T or u - T property diagram lines of constant pressure or constant volume respectively.

These procedures are applicable to solids, liquids and gases when particular pure substance exists in a single phase. Two or three-phase mixtures need a different approach. For liquids and solids the difference between Cp and Cv is very small because the change in volume with temperature during a constant pressure process is negligible compared to a gas.

When p is a constant and dv is negligible we can write:

dh = du |

The Perfect and Semi-Perfect Gases

The ideal gas is a gas which satisfies the relation P v = R T and its internal energy is a function of temperature alone i.e.

U = f(T), T is in Kelvin scale

We shall prove this later but accept it for the moment. Therefore

So that:

du= Cv dt

So for **an ideal gas**: u = f(T) then Cv must also be a function of temperature or a constant.

When Cv is a function of temperature we have a semi-perfect gas and when Cv is a constant we have a perfect gas.

i.e. U2 - U1 = Cv (T2 - T1) for a perfect gas.

**The Relation Between C****p****, C****v**** ****And R**

From the definition of enthalpy we have:

h = u + pv

= u + RT

R is the gas constant so we get:

dh = du + RdT

This relationship applies to both perfect and semi-perfect gases. So even when Cp and Cv vary with temperature their difference is always constant. Of course if Cv is a constant then Cp must be a constant too.

Enthalpy Change For A Perfect Gas

Now:

Hence:

## Universal or molar gas constant

Those of you who have done chemistry are aware that it is convenient to define an amount of substance called **mole:**

**The mole is the amount of substance which contains as many molecules as there are carbon atoms in 12 grams of carbon - 12**.

Mole is not a unit of mass but a quantity of substance corresponding to about 6.023 x10^6 particles or molecules.

The molar mass, M, of a substance in a system containing n moles and mass m is given by:

Consider 1 kmole of each of two gases (1) and (2) having molar masses M1 and M2. Then at a given p and T:

But Avagadro's hypothesis says that:

**"Equal volumes of gases at equal temperatures and pressures contain equal number of molecules"**

i.e. MR is the same for all ideal gases. It is known as the molar (universal) gas constant

The volume occupied by 1 kmol of an ideal gas at given p and T is fixed by Avagadro's hypothesis. At S.T.P this is 22.414 m3.

## Steam Tables and thermodynamic and transport properties of fluids

The steam tables used in the thermodynamics course are a set abstracted from more complete data published by the National Engineering Laboratory. They are designed to illustrate one convenient method for presenting the thermodynamic properties of a pure substance in equilibrium, when simple analytical relationships are not available. Where greater accuracy is required, the original tables should be used. It should be emphasised that there is little data on the properties of the solid phase since this is not normally of great importance in systems studied in engineering thermodynamics. Note that the units and notation are described inside the front cover. **Saturated water and steam**

These tables give the thermodynamic properties for those states where the liquid and vapour phases can exist together in equilibrium (as shown in image below). Pressure and temperature cannot be varied independently in the two phase region and because of the shape of the curve it is convenient to use temperature as the independent variable at low temperatures up to 100 deg C. For higher temperatures pressure is a more convenient variable.It is found that all states in this region can be represented in terms of the properties of saturated liquid and saturated vapour (subscripts f and g respectively). The difference between the values for water and steam is given the double subscript fg. Saturated liquid is defined as liquid, at a given pressure or temperature, for which any increase in its internal energy, enthalpy or volume must be accompanied by the formation of some vapour. Similarly saturated vapour (often called dry saturated vapour to emphasise the absence of any liquid). States intermediate between saturated water and saturated steam are often referred to as wet steam.

States intermediate between saturated liquid and vapour are conveniently defined in terms of a property, called the dryness fraction or quality, which is the raito of the mass of vapour to the total mass.

Thus dryness fraction x = m_{g}/(m_{f} + m_{g}) The use of the dryness fraction is illustrated by considering some steam at pressure p and a specific volume v which lies between the values v_{f} and v_{g} listed under pressure p. Dryness fraction x = (v-v_{f})/(v_{g}-v_{f})Enthalpy = h_{f} + x(h_{g} -h_{f})= h_{f} + x(h_{fg}) (A)= h_{g} - (1-x)h_{f} (B)=xh_{g} + (1-x)h_{f} These equivalent expressions can each be used with advantage in certain situations, expressions A and B being particularly appropriate for low and high values of the dryness fraction, respectively. **Superheated vapour** The data for superheated steam is presented using pressure and temperature as the independent variables for a limited number of pressures up to 1000 bar and certain temperatures up to 800 deg C. In general sufficient data is given to allow linear interpolation without too great an error although the data for high temperatures and pressures may not be entirely satisfactory. Where no data are recorded in these tables, it signifies that the state corresponding to this pressure and temperature is outside the superheated region, where values are required for interpolation in these regions use the values for dry saturated steam.

**Compressed liquid**

Compressed water is water at any temperature t above the freezing point and a pressure p, greater than the saturation pressure P_{sat} corresponding to the temperature t. it is found for compressed water that the variations in the extensive properties with change of pressure are small compared with the variations due to changes of temperature. In addition, lines of constant temperature are very nearly parallel straight lines. Therefore, it is convenient to tabulate, for different pressures, the changes in the extensive properties from their saturation values, corresponding to the temperature t. The table contains values of v - v_{f}, h - h_{f}, and s - s_{f} for a few values of pressure and temperature and linear interpolation is normally sufficiently accurate.Note: It is important to realise that the values listed show changes from the values v_{f} etc. corresponding to temperature t and pressure P_{sat}, and not the difference from the values v_{f} etc. corresponding to pressure p and temperature t'. (From the image above). **Triple point**

Under certain fixed conditions of pressure and temperature, the solid, liquid and vapour phases can exist together in equilibrium, the states for which such equilibrium exists are known as triple point states. If two extensive properties of a substance are plotted against each other, the triple point states define an area on the diagram, and a simple geometrical counstruciton allows the masses of the different phases to be determined. Thus on the u-v diagram (below) states S, L and V represent solid, liquid and vapour phases which can exist in equilibrium.

States lying on a line such as SB have a mass ratio of vapour to liquid given by the ratio of the lengths LB to BV. Hence state M has the following mass ratios:Mass of vapour/mass of solid = SC/CVMass of liquid/mass of solid = SA/ALMass of vapour/mass of liquid = LB/BV Knowing the values r_{s}, r_{L} and r_{v} of extensive property r at points S, L and V respectively, the value at M is calculated as follows: M x r = m_{s}r_{s} + m_{L}r_{L} + m_{v}r_{v} Where m_{s} is the mass of the solid phase present, etc. and m is the sum of m_{s}, m_{L} and m_{v}.