Real Gases and their Deviation from Ideality PDF

1.0 REAL GASES AND ITS DEVIATION FROM IDEALITY. 1.1 IDEAL GAS AND REAL GAS Before we discus real gases and its deviation from ideality we need to first remind ourselves about the key concepts of an ideal gas. What do we mean by an ideal gas? An ideal gas, also known as a perfect gas, is a hypothet...

1.0 REAL GASES AND ITS DEVIATION FROM IDEALITY. 1.1 IDEAL GAS AND REAL GAS Before we discus real gases and its deviation from ideality we need to first remind ourselves about the key concepts of an ideal gas. What do we mean by an ideal gas? An ideal gas, also known as a perfect gas, is a hypothetical concept in chemistry and physics used to simplify the study of gases. It is defined as: 1. Obedience to Gas Laws: An ideal gas is the gas that strictly obeys Boyle’s Law (P∝1/V at constant T), Charles’s Law (V∝T at constant P), and other classical gas laws under all conditions of temperature and pressure. 2. Obedience to the Gas Equation: It is a gas that follows the equation: PV=nRT where: o P= Pressure of the gas o V = Volume of the gas o n = Number of moles of the gas o R = Universal gas constant o T = Temperature in Kelvin Real Gas: Definition and Behavior A real gas is a gas that follows the gas laws only approximately under certain conditions. Unlike an ideal gas, real gases exhibit deviations from the ideal gas behavior, especially when subject to changes in pressure and temperature. The key points to note about real gases are:  A real gas is one that obeys gas laws reasonably well when the pressure is low or the temperature is high. Under these conditions, the gas molecules are far apart, and the effects of intermolecular forces and the finite size of gas particles are minimal, making the gas behave similarly to an ideal gas. www.testok.com.ng 1   In nature, there are no perfect or ideal gases. All gases, such as oxygen, nitrogen, and carbon dioxide, are examples of real gases.  As the pressure increases, the molecules of a real gas are forced closer together. This increases the impact of intermolecular forces (either attractive or repulsive), leading to behavior that deviates from the predictions of the gas laws.  Similarly, when the temperature decreases, the kinetic energy of the gas molecules decreases. At lower temperatures, the intermolecular forces become more significant, causing further deviations from ideal behavior. In summary, all gases are real gases, and their behavior depends on pressure and temperature conditions. The concept of real gases helps in understanding how molecular interactions affect gas properties. By recognizing these deviations, scientists can develop more accurate models for predicting gas behavior under various conditions, such as the Van der Waals equation. 1.2 Deviation of Real Gases from Ideal Behavior. According to Boyle’s law, the product of pressure and volume (PV) is constant when the temperature remains unchanged. This suggests that plotting PV against pressure (P) should yield a horizontal line parallel to the X-axis, indicating no variation in the PV value as pressure changes. However, practical observations show that real gases do not follow this ideal behavior exactly. Instead of a straight-line plot, two different curves are obtained due to the influence of intermolecular forces and the finite size of gas molecules. At low pressures, gases may approximate ideal behavior, but as pressure increases or temperature decreases, deviations become more pronounced. These deviations highlight the limitations of ideal gas assumptions in describing real gas behavior. (a) For gases like CO and CH4, the product PV first decreases with increase of pressure, reaches a minimum value and then begins to increase (b) For gases like hydrogen and helium, the product PV continuously increases with increases in Pressure. Fig1 2 1.3 Deviation of Real Gases from Boyle’s & Charles’ Law in Terms of Compressibility Factor,Z The effect of temperature and pressure on the behavior of a gas may be studied in terms of quantity ‘z’ called compressibility factor, which is defined for an ideal gas be one (z = 1) at all temperature and pressures. In case of real gases, the factor z varies from values less than 1 to values greater than 1, with changes of temperature and pressure. For example, the plots of compressibility factor z vs. pressure P at different constant temperatures for nitrogen gas are shown in fig 2. From these plots, it is observed that as the temperature increases, the minimum in the curves shift upwards. Ultimately, a temperature is reached at which the value of z remains close to 1 over an appreciable range of pressure. For example, in case of nitrogen gas, at 50°C, the value of z remains close to 1 up to nearly 100 atmospheres. This temperature at which a real gas behaves like an ideal gas over an appreciable pressure range is called Boyle temperature or Boyle point because at this temperature Boyle's law is obeyed over a range of pressures. Obviously, above the Boyle temperature, a gas shows positive deviations only. These deviations can be observed using the compressibility factor (Z), where: PV Z= nRT Fig 2 Deviations from Charles Gay-Lussac’s Law Charles Gay-Lussac’s law states that the volume of a gas is directly proportional to its absolute temperature when the pressure is kept constant. Mathematically, V = CONSTANT T V1 V2 = T1 T2 3 This law leads to the conclusion that the coefficient of volume expansion for all gases should be the same, regardless of the type of gas or the pressure applied. The coefficient of volume expansion is defined as the fractional increase in volume per degree Celsius rise in temperature, given by: 1 α= 127 per degree ceisus This means that for every 1°C rise in temperature, the volume of a gas should increase by 1/127 of its volume at 0°C. However, experimental observations show that this ideal behavior is not universally valid. The relationship holds true only at low pressures. At high pressures, gases exhibit significant deviations from the predictions of Charles Gay-Lussac’s law. These deviations illustrate the limitations of assuming gases behave ideally, especially under extreme conditions. Understanding these differences helps refine gas laws for real-world applications, where corrections like the Van der Waals equation are used to describe gas behavior more accurately. www.testok.com.ng Explanation of Deviations  At low pressures, gas molecules are far apart, and the influence of intermolecular attractions or repulsions is minimal. Under these conditions, gases follow Charles Gay- Lussac’s law closely.  As pressure increases, molecules are forced closer together, causing the volume change with temperature to differ from the ideal prediction. The gas may expand less than expected due to the intermolecular forces opposing the increase in volume. In summary, while Charles Gay-Lussac’s law provides a foundational model for understanding the relationship between temperature and volume, its assumptions are only valid at low pressures. Real gases require more comprehensive models to account for deviations observed at higher pressures. Example 19 Calculate the volume of 10 moles of methane at 100 atm pressure and 0°C. At this temperature and pressure, Z = 0.75. 4 1.4 Causes of Deviation of Real Gas from Ideal Behaviour These deviations occur because real gases do not behave ideally due to the effects of intermolecular forces and molecular size, which become more prominent as pressure increases. The gas laws are derived from kinetic theory of gases which is based upon certain assumptions. Thus there must be something wrong with certain assumptions. A careful study shows that at high pressure or low temperature, the following two assumptions of kinetic theory of gases become invalid. 1. The volume occupied by the gas molecules is negligible as compared to the total volume of the gas. 2. The forces of attraction or repulsion between the gas molecules are negligible. The above two assumptions are valid only if the pressure is low or the temperature is high so that the distance between the molecules is large. However, if the pressure is high or the temperature is low, the gas molecules come close together. Under these conditions: 1. The forces of attraction or repulsion between the molecules are not negligible. 2. The volume of the gas is so small that the volume occupied by the gas molecules cannot be neglected. 1.5 Derivation of van der Waals Equation for Real Gases Johannes Diderik van der Waals, a Dutch physicist, made a significant breakthrough in understanding the behavior of real gases in 1873. During his time, the ideal gas law (PV=nRT) was widely used, but it failed to describe gases accurately under high pressure and low temperature. Van der Waals was determined to solve this problem. He realized that real gases deviate from ideal behavior due to two key factors: 1. Intermolecular Forces: Unlike ideal gas molecules, real gas molecules attract each other. These attractions reduce the pressure exerted by the gas on its container walls. 2. Finite Molecular Volume: Real gas molecules occupy space, unlike the point-like particles assumed in ideal gases. This reduces the volume available for movement. To account for these factors, van der Waals modified the ideal gas law and introduced his famous van der Waals equation: 5  𝑎 (P+𝑉 2 ) (V-b) = RT where:  P = pressure,  V = molar volume,  T= temperature,  a = a constant accounting for intermolecular forces,  b = a constant accounting for the finite size of molecules,  R = universal gas constant. Van der Waals obtained the equation for real gases by applying the corrections for volume and pressure as explained below: 1. Correction for the Volume Suppose u is the actual volume occupied by one mole of the gas molecules. Then since the gas molecules are in motion, it has been found that the effective volume occupied by the gas molecules is four times the actual volume i.e., equal to 4 U. Let it be represented by b. The volume b is also called co-volume or excluded volume. Thus, the free space available for the movement of the gas molecules is (V- b). Hence, in the real gas equation, V should be replaced by (V -b) Fig 3 2. Correction for the Forces of Attraction, i.e. Pressure Correction. A gas molecule lying in the interior of the gas such as molecule A in Fig. 4 is attracted by all other gas molecules surrounding it. Hence, the net force of attraction exerted on such a molecule by the other molecules is zero. However, for a molecule lying near the wall of the container, such as molecule B in Fig. 4 the molecules lying inside the bulk of the gas exert some net inward pull. Thus, the effect of such an inward pull is “dragging back” of the molecule. Consequently, the pressure with which the gas molecule strikes the wall of the vessel is less than the pressure that would have been exerted if there were no such inward pull. Thus, the ideal pressure would be greater than the observed pressure by a factor p where p is the “correction factor due to the inward pull. This inward pull on the gas molecules lying near the wall depends upon (a) Number of molecules surrounding a molecule. (b) Total number of molecules Each of these factors in turn depends upon the density (ρ) of the gas. Hence, correction factor, 6 P α ρ2 But for a given mass of the gas, ρ α 1/V where V is the volume of the gas. Therefore, P α 1/V2 P= a/V2 Where a is a constant, depending upon the nature of the gas. Thus, corrected pressure = P + a/V2 Putting the corrected values of volume and pressure in the ideal gas equation, PV = RT, for 1 mole of the gas, the equation is modified to 𝑎 (P+𝑉 2 ) (V-b) = RT This equation is known as van der Waals equation. The constants a and b are called van der Waals constant whose values depend upon the nature of the gas and are independent of the temperature and pressure. For n moles of the gas, the corrected pressure become P+an2/ V2 and the effective volume = V- nb. Hence, the van der Waals equation takes the form 𝑎𝑛2 (P+ 𝑉 2 ) (V-nb) =nRT www.testok.com.ng 1.6 Significance of van Der Waals Constants a and b It is found that the values of a for the easily liquefiable gases are greater than those for the so- called permanent gases like H2 and He. Moreover, the value of a increases with the ease of liquefaction of the gas. We know that a gas which is more easily liquefiable has greater intermolecular forces of attraction. Hence, ‘a’, is a measure of the intermolecular forces of attraction in a gas, *b' is the ‘effective volume’ of the gas molecules. The constancy in the value of b for any gas over a wide range of temperature and pressure confirms that the gas molecules are incompressible. 1.7 Behaviour of Real Gases Using van Der Waals Equation 1. A Low Pressure At extremely low pressure, V is very large. Hence, the correction term a/V2 is very small. Similarly, the correction term b is also very small as compared to V. Thus, both the 7 correction terms can be neglected so that the van der Waals equation reduces to PV = RT. This is why at extremely low pressure, the gases obey the ideal gas equation. 2. At Moderate Pressure As the pressure is increased, the volume decreases and hence the factor a/V2 increases. Thus, the factor a/V2 can no longer be neglected. However, if the pressure is not too high, the volume V is still sufficiently large so that b can be neglected in comparison with V. Thus, the van der Waals equation reduces to 𝑎 (P+𝑉 2 ) V = RT 𝑎 PV+ = RT 𝑉 𝑎 PV= RT – 𝑉 Thus, PV is less than RT by a factor a/V. As the pressure increases, V decreases, so that the factor a/V increases. Thus, PV decreases as the pressure is increased. This explains why a dip in the plots of PV vs P of real gases is obtained (see Fig. 1) 3. At High Pressure As the pressure is increased further so that it is fairly high, V is so small that b can no longer be neglected in comparison with V. Although under these conditions, the factor a/V2 is quite large but since P is very high so a/V2 can be neglected in comparison with P. Thus, the van der Waals equation reduces to P(V-b) = RT PV - Pb = RT PV = RT + Pb Thus, PV is greater than RT by a factor Pb. Now as the pressure is increased, the factor Pb increases more and more. This explains why after the minima in the curves, the product PV increases continuously as the pressure is increased more and more (Fig. 1). 4. At High Temperature At any given pressure, if the temperature is sufficiently high, V is very large so that as in case 1, the van der Waals equation reduces to PV = RT. Hence, at high temperature, real gas behaves like an ideal gas. 8 1.8 Exceptional Behaviour of Hydrogen and Helium In case of hydrogen and helium, their molecules have very small masses. Hence, in the case of these gases, intermolecular forces of attraction are extremely small even at high pressures. In other words, the factor a/V is negligible at all pressures. Hence, the van der Waals equation is applicable in the following form at all pressures and ordinary temperatures. P (V -b) = RT or PV = RT + Pb This explains why hydrogen and helium show positive deviation only which increases with increase in the value of P. Example 2. Two moles of ammonia gas are enclosed in a vessel of 5-litre capacity at 27°C. Calculate the pressure exerted by the gas. Assuming that (a) The gas behaves like an ideal gas (using ideal gas equation) (b) The gas behaves like a real gas (using van der Waals’ equation) Given that for ammonia, a = 4.17 atm litre2 mol-2 (or 4.17 atm dm6 mol-2) and b = 0.037 litre mol- 1 (or 0.037 dm3 mol-1) Exercise 1 Calculate the pressure exerted by one mole of carbon dioxide gas in a 1.32 dm3 vessel at 48°C using the van der Waals equation. The van der Waals’constants are a=3.59 atm dm6 mol- 2 and b = 0.0427 dm3 mol-1 Exercise 2. Calculate from the van der Waals equation the temperature at which 3 moles of SO 2 would occupy a volume of 0.01m3 at 1519875 Nm-2 pressure (a = 0.679 Nm4 mol-2, b =5.64 x10-5 m3 mor1, R = 8.314 JK-1mol-1) 1.9 Kammerlingh-Onnes (Virial) Equation of State for Real Gases The Kammerlingh-Onnes Virial Equation of State is a thermodynamic model used to describe the behavior of real gases, taking into account the deviations from ideal gas behavior by introducing a series expansion. The equation expresses the compressibility factor (Z) of a gas as a power series in terms of the molar volume or in terms of pressure. Recall that PV Z= nRT For 1 mole of gas 9  PV Z= RT PV = ZRT PV = A + BP + CP2 + DP3 +... PV = RT ( 1 + BP + CP2 + DP3 +... ) in terms of pressure PV = A + B/V + C/V2 + D/ V3 +… PV = RT (1 + B/V + C/V2 + D/ V3 + … ) in terms of volume Where P is the pressure in atmospheres and V is the molar volume in litres. The coefficients A, B, C etc., are known respectively as the first, second, third, etc., virial coefficients. The following points may be noted about the above equation: 1. The virial coefficients are different for different gases. 2. At very low pressure, only the first virial coefficient is significant and it is equal to RT, i.e. A= RT 3. At higher pressures, the other virial coefficients also become important and must also be considered. 4. For any particular gas, the values of A, B, C, etc., are constant at constant temperature. Their values change with change of temperature. The first virial coefficient A is always positive and increases with temperature. On the other hand, the second virial coefficient B is negative at low temperatures. With increase of temperature, it increases to zero and then becomes more and more positive. Example 3: What is the molar volume of N2 (g) at 500 K and 600 atm according to (a) the perfect gas law, and (b) the virial equation? The virial coefficient B of N2 (g) at 500 K is 0.0169 litre mol. 1.10 BOYLE’S TEMPERATURE Boyle's temperature is the temperature at which a real gas behaves most like an ideal gas over a range of pressures. At this temperature, the gas's compressibility factor (Z) is equal to 1 for a range of pressures, meaning the effects of intermolecular forces are minimized. TB= a/Rb www.testok.com.ng 1.11 Second Virial Coefficient using van der Waals Equation The general equation of state as given by Onnes for the volume form is as follows: 𝐵 𝐶 PV = RT [1 + 𝑉 + +⋯] EQ3.1 𝑉2 10 The second virial coefficient B of this equation can be obtained by comparing it with van der Waals equation. 𝑎 (P+𝑉 2 ) (V-b) = RT 𝑎 𝑎𝑏 PV- Pb + 𝑉 − 𝑉 2 = 𝑅𝑇 Which may be re-written as 𝑎 𝑎𝑏 PV= 𝑅𝑇 + Pb - 𝑉 + 𝑉2 𝑎 As a and b are very small and if the pressure is not too high, 𝑉 2 can be neglected. Further, P may 𝑅𝑇 be replaced by 𝑉 Hence we get: 𝑅𝑇 𝑎 PV= 𝑅𝑇 + b -𝑉 𝑉 1 𝑎 PV= 𝑅𝑇 [1+ 𝑉(b - 𝑅𝑇)] EQ3.2 Comparing Equation (3.1) and (3.64), we get 𝑎 B= b - 𝑅𝑇 1.12 Other Equations of State Besides van der Waals equations and virial equation, a number of other equations have been worked out to give P-V-T relation in case of real gases. Some of these are described below. In all these gases, the quantity of gas considered is one mole. 1. The Berthelot Equation: is an empirical modification of the van der Waals equation of state for real gases, proposed by D. Berthelot in 1907. It aims to improve accuracy in describing the 11 behavior of gases over a wider range of temperatures and pressures by adjusting the temperature dependence of the attraction parameter. 𝑎 (P+𝑇𝑉 2 ) (V-b) = RT 𝑎 𝑎 Here, the correction term for pressure is in place of. 𝑇𝑉 2 𝑉2 2. Dieterici Equation: is an empirical model describing the behavior of real gases, similar to the van der Waals equation, but with a different approach to incorporating intermolecular forces. It was proposed by Conrad Dieterici in 1899. 𝑅𝑇 P = 𝑉−𝑏 𝑒 −𝑎/𝑅𝑇𝑉 This equation gives satisfactory results at moderate pressure in line with the van der Waals equation. However, at higher pressures, it gives results in greater agreement with experimented data. 3. Radlich-Kwong Equation: The Redlich-Kwong equation of state is an improved model for describing the behavior of real gases. It refines the Van der Waals equation by providing better accuracy at higher temperatures and pressures. This equation may be written as 𝑎 (P+𝑉(𝑉+𝑏)𝑇 1/2) (V-b) = RT This equation has been found to be the best two-parameter equation of state for real gases. 4. Clausius Equation: To account for the variation of a with temperature, Clausius used the following equation: 𝑎 (P+𝑇(𝑉+𝐶)2) (V-b) = RT. www.testok.com.ng 12