Chapter 2: Pure Substances

b) The Ideal Gas Equation of State

We continue with our discussion on Pure Substances.

We find that for a pure substance in the superheated region, at specific volumes much higher than that at the critical point, the P-v-T relation can be conveniently expressed by the Ideal Gas Equation of State to a high degree of accuracy, as follows:

P v = R T

where: R is constant for a particular substance and is called the Gas Constant

Note that for the ideal gas equation both the pressure P and the temperature T must be expressed in absolute quantities.

Consider for example the T-v diagram for water as shown below:

The shaded zone in the diagram indicates the region that can be represented by the Ideal Gas equation to an error of less than 1%. Note that at the critical point the error is 330%.

The gas constant R can be expressed as follows:

The three commonly used formats to express the Ideal Gas Equation of State are:

Solved Problem 2.5 - A piston-cylinder device contains 0.5 kg saturated liquid water at a pressure of 200 kPa. Heat is added and the steam expands at constant pressure until it reaches 300 °C .

• a) Draw a diagram representing the process showing the initial and final states of the system.

• b) Sketch this process on a T-v (temperature-specific volume) diagram with respect to the saturation lines, critical point, and relevant constant pressure lines, clearly indicating the initial and final states.

• c) Using steam tables determine the initial temperature of the steam prior to heating.

• d) Using steam tables determine the final volume of the steam after heating

• e) Using the ideal gas equation of state determine the final volume of the steam after heating. Determine the percentage error of using this method compared to that of using the steam tables.

Note: The critical point data and the ideal gas constant for steam can be found on the first page of the steam tables .

Solution Approach:

Even if questions a) and b) were not required, this should always be the first priority item in solving a thermodynamic problem.

c) Since state (1) is specified as saturated liquid at 200 kPa, we use the saturated pressure steam tables to determine that T ₁ = T _{sat@ 200kPa} = 120.2°C.

d) From the T-v diagram we determine that state (2) is in the superheated region, thus we use the superheated steam tables to determine that v ₂ = v _{200kPa,300°C} = 1.3162 m ³ /kg. Thus V ₂ = m,v ₂ = (0,5kg).(1.3162 m ³ /kg) = 0.658 m ³ (658 liters)

Note that in doing a units check we find that the following conversion appears so often that we feel it should be added to our Units Conversion Survival Kit (recall Chapter 1 ):

Finally we determine the percentage error of using the ideal gas equation at state (2)

Problem 2.6 - Consider a rigid container having a volume of 100 liters, filled with steam at an initial state of 400 kPa and 300°C. The steam is then cooled until it reaches a temperature of 90°C.

• a) Draw a diagram representing the process showing the initial and final states of the system.

• b) Using steam tables determine the mass of steam in the container. [0.153 kg]

• c) Using the ideal gas equation of state determine the mass of steam in the container. [0.151 kg]
  Determine the percentage error of using this method compared to that of using the steam tables. [1%]

• d) Sketch this process on a T-v (temperature-specific volume) diagram with respect to the saturation lines, critical point, and relevant constant pressure lines, clearly indicating the initial and final states.

• e) Using steam tables determine the final pressure and quality of the fluid mixture after cooling. [70.2 kPa, X = 0.277]

Note: The critical point data and ideal gas constant for steam can be found on the first page of the steam tables .

Solved Problem 2.7 - An automobile tire with a volume of 100 liters is inflated to a gage pressure of 210 kPa. Determine a) the mass of air in the tire if the temperature is 20°C, and b) the increase in gage pressure if the temperature in the tire reaches 50°C. Assume that atmospheric pressure is 100 kPa.

Solution Approach:

We always begin a thermodynamic problem with a sketch, indicating all the relevant information on the sketch, thus:

For part b) the temperature in the tire increases to 50°C (323K), however the volume and mass of air in the tire remains constant, thus:

(Note for the SI challenged - initially the pressure was 30 psig, and then rose to 35 psig. Try to validate these values)

Solved Problem 2.8 - In aircraft design it is common practice to specify a standard temperature distribution in the atmosphere near the surface of the earth (up to an elevation z of 10000m) as T(z) = (T ₀ + a.z)°C, where T ₀ at the earth's surface is 15°C, and a is the Temperature Lapse Rate (= - 0.00651°C / m). Using the Ideal Gas Equation of State, determine the pressure at an elevation of 3000m if at z = 0, P = 101 kPa.

Solution Approach:

We first develop the solution in terms of the Hydrostatic Equation on an elemental height of the column of air, the Ideal Gas Equation of State, and the Temperature Lapse Rate equation, as follows:

Solved Problem 2.9 - You may wonder why we would be interested in knowing the value of air pressure at 3000m altitude. In the following example we continue with the above development in order to evaluate the payload that can be lifted to an altitude of 3000m using a small hot air balloon (Volume =1000 m ³ ) having an empty mass of 100kg. Assume that the temperature of the air in the balloon is 100°C.

Solution Approach:

In this case we develop the solution in terms of a force balance between the bouyancy force (weight of the displaced air) and the gravity force including the weight of the hot air, the balloon empty mass, and the payload mass. The maximum altitude occurs when those two forces are equal, as follows:

Finally - with 154 kg payload at least 2 persons can share and enjoy this wonderful experience. Unfortunately they will not be able to enjoy a decent cup of English tea. At a pressure of 69.9 kPa water will boil at (heavens forbid) < 90 °C! (Saturation temperature T _sat from the Steam Tables ). Quick quiz: determine the temperature of a cup of tea in Denver, Colorado (elevation 5000 ft), or on the peaks of the Rocky Mountains (elevation 14000 ft. Hint : use the Units Survival Kit presented in Chapter1 to first convert from feet to meters)

Non-Ideal Gas Behavior

We noticed in the above T-v diagram for water that the gasses can deviate significantly from the ideal gas equation of state in regions nearby the critical point and there have been many equations of state recommended for use to account for this non-ideal behaviour. However, this non-ideal behaviour can be accounted for by a correction factor called the Compressibility Factor Z defined as follows:

thus when the compressibility factor Z approaches 1 the gas behaves as an ideal gas. Note that under the same conditions of temperature and pressure, the compressibility factor can be expressed as:

Different fluids have different values of critical point pressure and temperature data P _CR and T _CR , and these can be determined from the Table of Critical Point Data of Various Substances . Fortunately the Principle of Corresponding States states that we can normalize the pressure and temperature values with the critical values as follows:

All fluids normalized in this manner exhibit similar non-ideal gas behaviour within a few percent, thus they can all be plotted on a Generalised Compressibility Chart. A number of these charts are available, however we prefer to use the Lee-Kesler (logarithmic) Compressibilty Chart , The use of the compressibility chart is shown in the following example.

Solved Problem 2.10 - Carbon Dioxide gas is stored in a 100 liter tank at 6 MPa and 30°C. Determine the mass of CO ₂ in the tank based on (a) values obtained from the CO ₂ tables of data, (b) the ideal gas equation of state, and (c) the generalized compressibility chart. Compare (b) and (c) to (a) and determine the percentage error in each case.

Solution Approach:

We first determine the Critical Point data for CO ₂ from the Table of Critical Point Data of Various Substances

After evaluating the Reduced Pressure and Reduced Temperature we plot them on the Generalized Compressibility Chart in order to determine the Compressibility Factor, as shown below

The actual value of specific volume v _a is obtained from the CO ₂ Superheat Tables

The general rule is that if P << P _CR or if T >> T _CR then you are probably dealing with an ideal gas. If in doubt always check the Compressibility Factor Z on the Compressibility Chart.

Engineering Thermodynamics by Israel Urieli is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License