Unit 5: Real Gases

1. Behaviour of Real Gases & Deviations from Ideal State

Ideal Gas vs. Real Gas

An ideal gas is a theoretical gas that strictly obeys the ideal gas equation (PV = nRT) under all conditions of temperature and pressure. This behavior is based on the Kinetic Theory of Gases, which assumes that gas molecules have negligible volume and do not exert any intermolecular forces of attraction on one another.

A real gas does not behave ideally. Real gas molecules have a finite volume and experience intermolecular attractive and repulsive forces. Consequently, real gases only approximate ideal behavior at high temperatures and low pressures, where the molecules are far apart and moving rapidly.

Deviations from the Ideal Gas Equation

The extent to which a real gas deviates from ideal behavior is quantitatively measured using the Compressibility Factor (Z):

Z = PV / (nRT)

For one mole of gas (n = 1):

Z = PV / (RT)

• For an Ideal Gas: Z = 1 at all temperatures and pressures.
• For Real Gases: Z deviates from 1, and its value depends on pressure and temperature.

Behavior of Z with Pressure

• At Low Pressures: Intermolecular forces of attraction pull the molecules closer together, making the gas more compressible than expected. Therefore, PV is less than RT, and Z < 1 (Negative Deviation).
• At High Pressures: Gas molecules are packed tightly together. The finite volume of the molecules themselves becomes significant, resisting further compression. Therefore, PV is greater than RT, and Z > 1 (Positive Deviation).
• At Very Low Pressures (approaching zero): The molecules are extremely far apart, intermolecular forces are negligible, and Z approaches 1. Real gases then behave ideally.

Comparison Table

Important Observation: For every real gas, there exists a specific temperature at which it obeys the ideal gas law over a wide range of low pressures. This temperature is called the Boyle Temperature (Tb).

2. Andrew's Experiments on CO2 Gas

In 1869, Thomas Andrews performed a series of systematic experiments on Carbon Dioxide (CO2) to study the transition between gaseous and liquid states. He plotted the Pressure-Volume (P-V) relationships at various constant temperatures (called isotherms).

Key Experimental Observations

• High Temperature Isotherms (e.g., 48.1 °C): At high temperatures, the isotherms closely resemble Boyle's law hyperbolas. The gas cannot be liquefied, no matter how much pressure is applied.
• Critical Temperature Isotherm (31.1 °C): As the gas is compressed, the curve becomes flatter. At exactly 31.1 °C, a point of inflection occurs where the curve has a horizontal tangent. This represents the critical point. Above this temperature, CO2 remains a gas.
• Low Temperature Isotherms (e.g., 21.5 °C and 13.1 °C): These isotherms display three distinct regions during compression:
  1. Gaseous Region: At low pressure, the volume decreases as pressure increases, following a curve.
  2. Horizontal Line (Coexistence Region): Condensation begins at a specific pressure. The pressure remains constant while the volume decreases sharply as gas transitions into liquid. In this horizontal region, liquid and vapor phases coexist in equilibrium.
  3. Liquid Region: Once the gas is entirely liquefied, further compression requires extremely high pressure to produce even a tiny change in volume. The curve becomes nearly vertical because liquids are highly incompressible.

Exam Note: The horizontal portion of the isotherms shortens as the temperature rises, eventually shrinking to a single point at the critical temperature (31.1 °C).

3. Critical Constants & Continuity of State

Definitions of Critical Constants

The state of a gas at its critical point is defined by three critical constants:

Critical Temperature (Tc): The maximum temperature at which a gas can be liquefied by the application of pressure alone. Above Tc, a substance cannot exist in the liquid state, regardless of the pressure applied. For CO2, Tc = 31.1 °C (304.2 K).

Critical Pressure (Pc): The minimum pressure required to liquefy a gas at its critical temperature. For CO2, Pc = 73 atmospheres (73.9 atm).

Critical Volume (Vc): The volume occupied by one mole of a gas at its critical temperature and critical pressure.

Continuity of Liquid and Gaseous States

Andrews' experiments demonstrated that the liquid and gaseous states are not separated by an absolute boundary, but rather represent two ends of a continuous state of matter. This is known as the Continuity of State.

It is possible to convert a gas into a liquid without undergoing a sudden, visible phase separation (such as condensation along a horizontal isotherm line). This continuous transition is achieved by:

1. Heating the gas above its critical temperature (Tc) at a low pressure.
2. Compressing the gas at this high temperature (where no liquefaction can occur) until its pressure exceeds the critical pressure (Pc).
3. Cooling the substance below its critical temperature (Tc) while keeping the pressure high.
4. Reducing the pressure. The substance will now behave as a liquid without ever having shown a distinct boundary or meniscus separating the liquid and gas phases.

Difference Between Vapour and Gas

4. Van der Waal's Equation of State

In 1873, J.D. van der Waals modified the ideal gas equation to account for the behavior of real gases by introducing two correction factors: one for volume and one for pressure.

1. Volume Correction (Co-volume / Excluded Volume)

Gas molecules are not mathematical points; they possess a finite physical volume. The actual space available for the free movement of molecules is less than the total volume of the container (V).

The excluded volume per mole of gas is represented by the constant b (also called the co-volume). For n moles, the excluded volume is nb. Therefore, the corrected volume is:

Corrected Volume = V - nb

The constant b is equal to four times the actual molecular volume of one mole of gas:

b = 4 * N * Vm

Where N is Avogadro's number and Vm is the actual volume of a single molecule.

2. Pressure Correction (Cohesive Forces)

A molecule inside the bulk of a gas experiences equal attractive forces from all directions, resulting in a net attractive force of zero. However, a molecule striking the container wall experiences a net inward pull from the molecules behind it. This inward pull reduces the force and speed with which the molecule strikes the wall, lowering the measured pressure (P).

To compensate, a pressure correction term (p) must be added to the measured pressure:

Corrected Pressure = P + p

This inward pull is proportional to:

• The number of molecules striking the wall per unit area (which is proportional to density, ρ).
• The number of molecules pulling them back from behind (also proportional to density, ρ).

Since density (ρ) is inversely proportional to the volume (V) of the gas:

p is proportional to ρ²

p is proportional to (1 / V²)

p = a / V² (for one mole)

p = an² / V² (for n moles)

Where a is the Van der Waals constant representing the strength of intermolecular attractive forces.

The Van der Waals Equation

For one mole of gas (n = 1):

(P + a / V²)(V - b) = RT

For n moles of gas:

(P + an² / V²)(V - nb) = nRT

Common Mistake: Ensure the units of the constants are correct. For one mole:

• a: atm L² mol⁻²
• b: L mol⁻¹

5. Values of Critical Constants & Isotherm Comparison

Derivation of Critical Constants in terms of Van der Waals Constants

At the critical point on a P-V diagram, the critical isotherm exhibits a point of inflection. At this point, the mathematical conditions are:

(dP / dV) = 0 and (d²P / dV²) = 0

Let us express the Van der Waals equation for one mole of gas as pressure:

P = RT / (V - b) - a / V²

Differentiating P with respect to V at constant temperature T:

(dP / dV) = -RT / (V - b)² + 2a / V³ = 0

This gives equation (1):

RT / (V - b)² = 2a / V³

Differentiating again with respect to V:

(d²P / dV²) = 2RT / (V - b)³ - 6a / V⁴ = 0

This gives equation (2):

2RT / (V - b)³ = 6a / V⁴

Dividing equation (2) by equation (1):

2 / (V - b) = 3 / V

2V = 3(V - b)

2V = 3V - 3b

V = 3b

At the critical point, the volume is the critical volume Vc:

Vc = 3b

Substitute Vc into equation (1) to find the critical temperature Tc:

RTc / (3b - b)² = 2a / (3b)³

RTc / 4b² = 2a / 27b³

Tc = 8a / (27 * R * b)

Now, substitute Vc and Tc back into the expression for pressure to find the critical pressure Pc:

Pc = RTc / (Vc - b) - a / Vc²

Pc = [ R * (8a / 27Rb) ] / (3b - b) - a / (3b)²

Pc = [ 8a / 27b ] / 2b - a / 9b²

Pc = 4a / 27b² - 3a / 27b²

Pc = a / 27b²

Critical Compressibility Factor (Zc)

The compressibility factor at the critical point is calculated as:

Zc = Pc * Vc / (R * Tc)

Substituting the derived critical constants:

Zc = (a / 27b² * 3b) / (R * 8a / 27Rb) = (3a / 27b) / (8a / 27b) = 3 / 8 = 0.375

Important Observation: Van der Waals equation predicts Zc = 0.375 for all gases. However, experimental values for most real gases are lower (typically around 0.28). This discrepancy shows that while the Van der Waals model is an excellent qualitative description, it has limitations in quantitative precision.

Comparison with Experimental Curves (P-V Diagrams)

• Experimental Isotherms (Andrews): Below the critical temperature, experimental isotherms show a flat, horizontal line representing the liquid-gas phase transition. In this region, pressure remains constant as volume decreases.
• Theoretical Isotherms (Van der Waals): The Van der Waals equation is a cubic equation in terms of volume (V). Below Tc, it yields a wave-like curve (a loop) with a local maximum and a local minimum instead of a horizontal line.
• The Van der Waals Loop: The parts of the loop that have positive slopes (where volume increases as pressure increases) represent physically unstable states that cannot exist in stable equilibrium. However, the portions of the loop near the local maximum and minimum can be observed under highly controlled, dust-free conditions as metastable states (supersaturated vapor and superheated liquid).

6. Law of Corresponding States

The Law of Corresponding States shows that if we express the state of a gas in terms of scaled variables relative to its critical constants, all real gases behave according to a single equation of state, regardless of their chemical identity.

Reduced Variables

We define the dimensionless reduced variables as:

• Reduced Pressure: Pr = P / Pc
• Reduced Volume: Vr = V / Vc
• Reduced Temperature: Tr = T / Tc

Derivation of the Reduced Equation of State

We substitute the expressions for actual pressure, volume, and temperature (P = Pr*Pc, V = Vr*Vc, T = Tr*Tc) into the Van der Waals equation for one mole of gas:

(P + a / V²)(V - b) = RT

(Pr * Pc + a / (Vr² * Vc²))(Vr * Vc - b) = R * Tr * Tc

Substitute the values of the critical constants (Pc = a / 27b², Vc = 3b, Tc = 8a / 27Rb):

(Pr * (a / 27b²) + a / (9b² * Vr²))(3b * Vr - b) = R * Tr * (8a / 27Rb)

Factor out common terms:

[ a / (27b²) * (Pr + 3 / Vr²) ] * [ b * (3Vr - 1) ] = 8a * Tr / (27b)

[ a / (27b) ] * (Pr + 3 / Vr²)(3Vr - 1) = [ 8a / (27b) ] * Tr

Dividing both sides by a / 27b yields the Reduced Equation of State:

(Pr + 3 / Vr²)(3Vr - 1) = 8Tr

Significance: This equation is completely free of the substance-specific constants a and b. It applies universally to all real gases. If two different gases have the same reduced temperature and reduced pressure, they will occupy the same reduced volume. Under these conditions, the gases are said to be in corresponding states.

7. Joule-Thomson Porous Plug Experiment & Effect

The Joule-Thomson effect describes the temperature change of a real gas when it expands through a barrier from a region of constant high pressure to a region of constant low pressure.

The Porous Plug Experiment

In 1852, James Prescott Joule and William Thomson (Lord Kelvin) designed an experiment to measure this effect:

• A continuous stream of gas at pressure P1 and temperature T1 was forced through a porous plug (such as cotton wool or silk) into a region of lower pressure P2.
• The system was enclosed in a thick, thermally insulating box to ensure the process was strictly adiabatic (no heat exchanged with the surroundings, so Q = 0).
• They measured the temperature of the gas before (T1) and after (T2) it passed through the plug.

Thermodynamic Proof of Constant Enthalpy

Let us consider one mole of gas passing through the porous plug:

• Work done on the gas to push it through the plug at the high-pressure side: W1 = -P1 * V1
• Work done by the gas as it expands on the low-pressure side: W2 = P2 * V2
• Net work done by the gas: W = P2 * V2 - P1 * V1

According to the First Law of Thermodynamics:

dU = Q - W

Since the process is adiabatic, Q = 0:

U2 - U1 = -(P2 * V2 - P1 * V1)

U2 + P2 * V2 = U1 + P1 * V1

Since Enthalpy (H) is defined as H = U + PV:

H2 = H1

Conclusion: The Joule-Thomson expansion is an isenthalpic process (enthalpy remains constant).

The Joule-Thomson Coefficient (μ)

The Joule-Thomson coefficient (μ) measures how much the temperature changes with a change in pressure at constant enthalpy:

μ = (dT / dP) at constant H

• If μ > 0 (Positive): A decrease in pressure (dP < 0) results in a decrease in temperature (dT < 0). The gas cools during expansion.
• If μ < 0 (Negative): A decrease in pressure (dP < 0) results in an increase in temperature (dT > 0). The gas heats up during expansion.
• If μ = 0 (Zero): The temperature does not change during expansion.

Joule-Thomson Effect for an Ideal Gas

For an ideal gas, the enthalpy depends only on temperature (since there are no intermolecular forces to overcome during expansion). Because enthalpy is constant during the expansion, the temperature must also remain constant:

μ = 0 (for an Ideal Gas)

An ideal gas experiences neither cooling nor heating during a Joule-Thomson expansion.

Joule-Thomson Effect for a Van der Waals Gas

Using thermodynamic relations, the Joule-Thomson coefficient can be expressed as:

μ = (1 / Cp) * [ T * (dV / dT) - V ]

Where Cp is the heat capacity at constant pressure. For a Van der Waals gas, evaluating the term [T * (dV / dT) - V] yields the approximation:

μ = (1 / Cp) * [ (2a / RT) - b ]

8. Temperature of Inversion, Cooling, & Regenerative Cooling

Temperature of Inversion (Ti)

The Temperature of Inversion (Ti) is the temperature at which the Joule-Thomson coefficient (μ) is exactly zero. At this temperature, a gas undergoing expansion shows neither a temperature increase nor decrease.

We can find the inversion temperature by setting μ = 0 in the Van der Waals expression:

2a / (R * Ti) - b = 0

Ti = 2a / (R * b)

Relationship with Critical Temperature

We can relate Ti to the critical temperature Tc (Tc = 8a / 27Rb):

Ti = (27 / 4) * Tc = 6.75 * Tc

Cooling vs. Heating Regions

Real-World Application (Hydrogen and Helium): At room temperature, Hydrogen and Helium have temperatures well above their inversion temperatures (Ti for H2 is -80 °C, and for He is -242 °C). Therefore, if they are expanded at room temperature, they will heat up instead of cooling down. To cool and liquefy these gases using the Joule-Thomson effect, they must first be pre-cooled below their inversion temperatures using external refrigerants (like liquid nitrogen).

Regenerative Cooling

A single Joule-Thomson expansion produces only a small drop in temperature. To reach the very low temperatures required to liquefy gases, we use Regenerative Cooling.

Regenerative cooling is a continuous, repetitive process where the gas cooled by a Joule-Thomson expansion is used to cool the incoming, compressed gas before it expands. The system works as follows:

1. Highly compressed gas is cooled initially using a standard refrigerant.
2. The compressed gas passes through the inner tube of a double-walled counter-current heat exchanger.
3. The gas expands through a valve or porous plug, undergoing Joule-Thomson cooling and dropping in temperature.
4. This cooled, low-pressure gas is routed back through the outer tube of the heat exchanger, cooling the incoming high-pressure gas.
5. This feedback loop ensures that the incoming gas arrives at the expansion valve progressively colder and colder.
6. This continuous cycle lowers the temperature further with each pass until the gas eventually liquefies.

This process is the core principle behind the Linde Process and Hampson Process used for liquefying air, oxygen, nitrogen, and hydrogen on an industrial scale.