Gas

The ideal gas model and when it fails.

Ideal Gas

An ideal gas is a theoretical concept in physics and chemistry that simplifies the behavior of gases to facilitate understanding and calculations. It is based on several assumptions:

1. Point Particles: The molecules of an ideal gas are considered point particles with no volume. This means that the volume of the gas particles themselves is negligible compared to the volume the gas occupies.

2. Elastic Collisions: Collisions between gas particles and between particles and the walls of the container are perfectly elastic. This means there is no loss of kinetic energy during collisions.

3. No Intermolecular Forces: Ideal gas particles do not exert any forces on each other, except during collisions. There are no attractions or repulsions between particles.

4. Random Motion: The gas particles are in constant, random motion, and their collisions lead to a uniform distribution of velocities (Maxwell-Boltzmann distribution).

5. Kinetic Energy and Temperature: The average kinetic energy of gas particles is directly proportional to the absolute temperature of the gas.

The ideal gas law, \(PV = nRT\), where \(n\) is the number of moles and \(R\) is the universal gas constant, describes the behavior of an ideal gas. This law is a combination of Boyle’s law, Charles’s law, and Avogadro’s law.

While no real gas perfectly fits the description of an ideal gas, many gases behave approximately ideally under a wide range of conditions, especially at low pressures and high temperatures where the volume of the gas particles is much less significant compared to the volume of the container, and intermolecular forces are weaker.

The relationship \(PV = kNT\) connects pressure (\(P\)), volume (\(V\)), the number of molecules (\(N\)), the Boltzmann constant (\(k\)), and temperature (\(T\)) in a gas. This equation is a form of the ideal gas law, reflecting the kinetic theory of gases. Here’s the breakdown:

1. Pressure and Molecular Impacts: Pressure in a gas arises from the force exerted by molecules colliding with the walls of the container. This force per unit area (pressure) is related to the change in momentum of the molecules upon impact. The more frequent and more forceful these collisions, the higher the pressure.

2. Temperature and Average Kinetic Energy: The temperature of a gas is a measure of the average kinetic energy (\(\frac{1}{2}mv^2\)) of the molecules. As temperature increases, the average speed of the molecules (\(v\)) increases, leading to more frequent and more forceful collisions, which in turn increases the pressure.

3. Connection between Temperature, Speed, and Pressure: Higher temperatures mean higher average speeds for the gas molecules due to their increased kinetic energy. This results in more frequent and more energetic collisions with the container walls, thus increasing the pressure. Mathematically, this is expressed as \(\frac{1}{2}mv^2 = \frac{3}{2}kT\) for a monatomic gas, linking the average kinetic energy directly to temperature.

4. Ideal Gas Law (PV = NkT): This equation encapsulates the relationships mentioned above. It shows that the product of pressure and volume is proportional to the total kinetic energy of all the molecules in the system, which is a function of the number of molecules and the temperature.

• \(P\) (Pressure) increases with more frequent/energetic collisions.
• \(V\) (Volume) provides the space in which the molecules are contained. More volume means fewer collisions per unit area, thus lower pressure.
• \(N\) (Number of molecules) affects how many collisions occur.
• \(T\) (Temperature) influences the speed and kinetic energy of the molecules, affecting their collisions’ frequency and force.
• \(k\) (Boltzmann constant) is a proportionality factor that relates energy at the molecular level to temperature.

By understanding these connections, you can see how changes in temperature influence molecular speeds and how this affects the pressure in a system, given a constant volume and number of molecules.

\(C_v\) and \(C_p\) are specific heat capacities at constant volume and constant pressure, respectively. They represent the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree Celsius (or one Kelvin) under constant volume or pressure.

• \(C_v\) (Specific Heat at Constant Volume): This is the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree while keeping the volume constant. For an ideal gas, \(C_v\) is related to the internal energy and does not consider work done by the system as the volume is constant.

• \(C_p\) (Specific Heat at Constant Pressure): This is the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree while keeping the pressure constant. For an ideal gas, \(C_p\) is related to the enthalpy and takes into account the work done by the system during expansion.

The relationship between \(C_p\) and \(C_v\) for an ideal gas is given by the Mayer’s relation: \(C_p - C_v = R\), where \(R\) is the ideal gas constant.

These specific heat capacities are crucial in thermodynamics for understanding and calculating the energy exchanges in thermal processes.

Quantum Theory of Gasses

The quantum theory of gases incorporates quantum mechanics to describe the behavior of gas particles, a significant enhancement over the classical ideal gas law. This theory is particularly crucial when dealing with low temperatures or high densities, where quantum effects become pronounced. Here are key aspects of the quantum theory of gases:

1. Quantization of Energy Levels: Unlike classical particles, which can have any amount of energy, quantum theory dictates that particles like atoms and molecules in a gas have quantized energy levels. This includes translational, rotational, vibrational, and electronic states.

2. Bose-Einstein and Fermi-Dirac Statistics: The quantum theory of gases distinguishes between two types of particles: bosons (particles with integer spin) and fermions (particles with half-integer spin). Bosons obey Bose-Einstein statistics, allowing multiple particles to occupy the same energy state. Fermions, following the Pauli exclusion principle, obey Fermi-Dirac statistics, where no two particles can occupy the same quantum state simultaneously. These statistics lead to different behaviors for gases composed of bosons or fermions, especially at low temperatures.

3. Bose-Einstein Condensation: At extremely low temperatures, bosons can condense into the lowest available quantum state, forming a Bose-Einstein condensate. This state of matter exhibits quantum phenomena on a macroscopic scale.

4. Degenerate Fermi Gas: At low temperatures or high densities, a fermion gas can become a degenerate Fermi gas. In this state, all low-energy quantum states are filled up to a certain energy level called the Fermi energy. This behavior is crucial for understanding the properties of electrons in metals and the structure of neutron stars.

5. Energy Distribution: Quantum statistics affect the distribution of particles among various energy states. The Maxwell-Boltzmann distribution, adequate for classical gases, is replaced by the Bose-Einstein or Fermi-Dirac distribution for quantum gases.

6. Partition Functions and Thermodynamics: Quantum theory modifies the calculation of partition functions, which are central to determining the thermodynamic properties of gases. These functions account for the discrete energy levels and statistical behavior of particles.

The quantum theory of gases provides a more accurate description of gas behavior under conditions where classical physics is insufficient, especially at low temperatures or high densities, revealing the fascinating interplay between quantum mechanics and statistical physics.

Bosons and fermions are two categories of elementary particles distinguished by their spin, a quantum mechanical property. The particles that constitute ordinary matter—electrons, protons, and neutrons—are all fermions. Electrons have a spin of 1/2, making them fermions, and protons and neutrons, which are made up of quarks (also fermions with spin 1/2), collectively behave as fermions with spin 1/2 as well. Their differing spins result in distinct statistical behaviors, fundamentally influencing how they behave and interact:

1. Bosons:
   • Spin: Bosons have integer spins (0, 1, 2, …).
   • Statistics: They obey Bose-Einstein statistics, allowing multiple bosons to occupy the same quantum state simultaneously.
   • Behavior: This characteristic leads to phenomena like Bose-Einstein condensation, where at low temperatures, a large number of bosons occupy the lowest energy state.
   • Examples: Photons (particles of light with spin 1), gluons (spin 1, carriers of the strong force), and the Higgs boson (spin 0) are all examples of bosons. Additionally, composite particles like mesons and certain atomic nuclei (like helium-4) can behave as bosons.
2. Fermions:
   • Spin: Fermions have half-integer spins (1/2, 3/2, 5/2, …).
   • Statistics: They obey Fermi-Dirac statistics and are subject to the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state within a quantum system.
   • Behavior: This leads to the “filling up” of energy levels in atoms, resulting in the complex structure of the periodic table and the behavior of electrons in metals.
   • Examples: Electrons, protons, and neutrons are all fermions with spin 1/2. Quarks, which make up protons and neutrons, are also fermions.

The distinction between bosons and fermions is fundamental to understanding the behavior of quantum systems and is a key concept in quantum field theory, which describes the fundamental forces of nature. In condensed matter physics, this distinction explains a variety of phenomena, from the superfluidity of liquid helium to the electrical conductivity of metals and semiconductors.

Failures

Real gases deviate from the ideal gas law \(PV = nRT\) due to the effects of intermolecular forces and the finite volume of the gas molecules, which are not accounted for in the ideal gas model. The deviations become significant at high pressures and low temperatures. Here’s how real gases differ:

1. Intermolecular Forces: Ideal gases assume no intermolecular attractions or repulsions, but real gases experience these forces. At high pressures (or low temperatures), attractive forces between molecules reduce the pressure exerted on the container walls compared to what would be expected for an ideal gas.

2. Finite Molecular Volume: Ideal gases assume that the volume of the gas molecules themselves is negligible compared to the container’s volume. However, at high pressures (or low volumes), the finite size of the molecules becomes significant, and the available volume for molecular motion is less than the total volume of the container.

The Van der Waals equation is a well-known model that corrects for these deviations by introducing two constants, \(a\) and \(b\):

\[\left(P + \frac{an^2}{V^2}\right) (V - nb) = nRT\]

• \(a\): Corrects for the attractive forces between molecules. The \(\frac{an^2}{V^2}\) term is added to the pressure to account for these forces.
• \(b\): Corrects for the volume occupied by the gas molecules. The \(nb\) term is subtracted from the volume to reflect the reduced volume available for molecular motion.

These corrections help the Van der Waals equation more accurately predict the behavior of real gases, especially under conditions where the ideal gas law fails.

Humid air

Humid air calculations are an example where the ideal gas law might not provide entirely accurate answers due to the interactions between different gas molecules, particularly water vapor, and air. In the context of humid air:

1. Non-Ideal Behavior: Humid air is a mixture of dry air and water vapor. While each component can often be approximated as an ideal gas, their interactions, especially under high humidity and varying temperature conditions, can lead to non-ideal behavior. For example, the presence of water vapor affects the heat capacity and density of air.

2. Partial Pressures: In humid air, the total pressure is the sum of the partial pressures of dry air and water vapor, as stated by Dalton’s Law of Partial Pressures. The ideal gas law can be applied to each component separately, but the interactions between them, particularly at high humidity levels, can cause deviations.

3. Impact on Calculations: The ideal gas law might not accurately predict the behavior of humid air in scenarios like:

   • High Humidity: At high levels of humidity, especially close to the saturation point, the interactions between water molecules become significant.
   • Temperature Changes: The capacity of air to hold water vapor changes with temperature, affecting humidity calculations. Condensation and evaporation, influenced by temperature, can cause deviations from ideal behavior.

4. Psychrometrics: This field studies the thermodynamic properties of moist air and often requires corrections to the ideal gas law to accurately describe processes like air conditioning, heating, ventilation, and refrigeration.

While the ideal gas law provides a good approximation in many scenarios, engineers and scientists working with HVAC systems, weather prediction, and other fields involving humid air often use more complex models or empirical data to account for the non-ideal interactions between air and water vapor.

Failure of ideal theory of gasses

Question. Can you give a specific situation involving high humidity and temperature changes where the theory of ideal gasses will give an answer that is materially wrong. I’d like an example with specific variable values showing two answers that are materially different.

ans

Let’s consider a situation involving the condensation of water vapor in air, a common scenario where high humidity and temperature changes can lead to significant deviations from ideal gas behavior.

Scenario: A sealed container with a volume of 1 cubic meter contains air at 30°C with a relative humidity of 100%. The air is cooled to 20°C. We want to calculate the amount of water vapor that will condense due to this temperature change.

For simplicity, let’s assume the container is initially at 1 atmosphere (101325 Pa).

1. Using Ideal Gas Law:

At 30°C (303.15 K), the saturation vapor pressure of water is about 4240 Pa.

• Initial moles of water vapor \(n_{\text{water, initial}}\) can be calculated using \(PV = nRT\).
  • \(n_{\text{water, initial}} = \frac{P_{\text{water}}V}{RT} = \frac{4240 \times 1}{8.314 \times 303.15}\) moles.

When the air cools to 20°C (293.15 K), the saturation vapor pressure of water drops to about 2338 Pa.

• The amount of water vapor the air can hold at 20°C is \(n_{\text{water, 20°C}} = \frac{2338 \times 1}{8.314 \times 293.15}\) moles.
• The difference in moles between the initial state and the final state will condense out.

2. Considering Non-ideal Behavior:

Real gases, especially near the condensation point, do not follow the ideal gas law precisely due to intermolecular forces. The actual amount of water that condenses can be influenced by:

• The non-ideal behavior of water vapor, especially as it approaches the dew point.
• The interaction between water molecules and air, which can affect the condensation process.

In reality, the amount of water condensing could be influenced by factors like the presence of condensation nuclei or surface interactions within the container, which are not accounted for by the ideal gas law.

Outcome:

The ideal gas law might predict a certain amount of water to condense based on purely thermodynamic calculations. However, in practice, the actual amount might differ due to the reasons mentioned above, such as the need for nucleation sites for condensation or variations in local conditions within the container that affect how water vapor condenses.

This example highlights how ideal gas calculations might not fully capture the nuances of phase changes and interactions in a real gas mixture, especially when dealing with high humidity and temperature changes that are critical for condensation processes.

Gemini example

Absolutely! Here’s a specific example focused on the density of air, where the ideal gas law can give a materially inaccurate result in humid conditions:

Scenario: Calculating the density of air for buoyancy considerations (like weather balloon design).

Conditions

• Air Temperature: 30°C (86°F)
• Pressure: Standard atmospheric pressure (101325 Pa)
• Case 1 (Dry Air): Relative Humidity = 0%
• Case 2 (Humid Air): Relative Humidity = 80%

Calculations

• Ideal Gas Law (Simplified for Density): ρ = (P * M) / (R * T) Where:

  • ρ = Density (kg/m³)
  • P = Pressure (Pa)
  • M = Molar Mass (kg/mol)
  • R = Universal Gas Constant (8.314 J/mol*K)
  • T = Temperature (Kelvin)

Case 1: Dry Air

• Molar Mass of Dry Air ≈ 0.02897 kg/mol
• T = 30°C + 273.15 = 303.15 K
• Calculation: ρ ≈ (101325 * 0.02897) / (8.314 * 303.15) ≈ 1.164 kg/m³

Case 2: Humid Air

• Water vapor is now significant. We need to adjust the average molar mass of the air mixture.
• Approximate Molar Mass of Water Vapor = 0.018 kg/mol
• With 80% relative humidity, a good approximation is that water vapor accounts for about 20% of the molecules in the air mixture.
• Effective Molar Mass ≈ (0.8 * 0.02897) + (0.2 * 0.018) ≈ 0.02678 kg/mol
• Calculation: ρ ≈ (101325 * 0.02678) / (8.314 * 303.15) ≈ 1.080 kg/m³

The Difference

The ideal gas law in this instance suggests dry air has a density of 1.164 kg/m³. However, in humid conditions, the actual density is closer to 1.080 kg/m³. This is about a 7% difference!

Implications

This density difference could significantly impact the buoyancy calculations of a weather balloon. Underestimating the density could lead to the balloon not having enough lift and failing to reach the desired altitude.

Important Note This demonstrates a material error due to neglecting humidity. For greater accuracy, equations beyond the simple ideal gas law or specialized psychrometric calculations should be used in situations involving humid air.

Adiabatic compression in a car cylinder

Question. Calculate the increase in temperature as air at 20C is compressed in an internal combustion engine at a 12:1 compression ratio. Assume that the compression is adiabatic.

ans

In an adiabatic process, there is no heat exchange with the surroundings, and the work done on the gas is converted into internal energy, increasing the temperature. The relationship between temperature and volume in an adiabatic process for an ideal gas is given by:

\[T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1}\]

Where:

• \(T_1\) and \(T_2\) are the initial and final temperatures in Kelvin.
• \(V_1\) and \(V_2\) are the initial and final volumes.
• \(\gamma\) (gamma) is the heat capacity ratio (\(C_p / C_v\)). For diatomic gases like air, \(\gamma\) is approximately 1.4.

Given: - The initial temperature \(T_1 = 20°C = 293.15\) K (since 0°C = 273.15 K). - The compression ratio \(V_1 / V_2 = 12:1\).

We can express \(V_2\) in terms of \(V_1\) as \(V_2 = V_1 / 12\). Now, we can solve for \(T_2\) using the adiabatic relationship:

\[T_1 V_1^{\gamma - 1} = T_2 V_2^{\gamma - 1}\] \[T_2 = T_1 \left( \frac{V_1}{V_2} \right)^{\gamma - 1}\] \[T_2 = 293.15 \times 12^{1.4 - 1}\]

Let’s calculate \(T_2\).

After the adiabatic compression, the final temperature \(T_2\) is approximately 792.07 K, or about 518.92°C. This significant increase in temperature during compression in an internal combustion engine is a critical factor in the engine’s efficiency and power generation.

ans 1

The increase in temperature during adiabatic compression is less than what you might expect from a simple 12x scaling suggested by the ideal gas law (\(PV = nRT\)) because the energy is not just going into increasing the linear velocities of the molecules (which would directly increase temperature) but also into doing work against the intermolecular forces as the molecules are compressed closer together.

When you compress a gas adiabatically:

1. Work Done on the Gas: The work done to compress the gas is converted into internal energy, increasing the temperature. However, this energy increase doesn’t translate directly into a proportional rise in temperature because some of the work is used in overcoming the attractive forces between molecules as they are pushed closer together, especially as the gas density increases.

2. Molecular Kinetic Energy: While the average kinetic energy of the molecules (which is related to temperature) does increase, the relationship is not linear. The increase in kinetic energy and hence temperature depends on the heat capacity ratio (\(\gamma\)), which reflects the degrees of freedom of the molecules. In a diatomic gas like air, some energy goes into rotational and vibrational modes, not just translational kinetic energy.

3. Non-ideal Gas Behavior: As the gas is compressed, it behaves less ideally. The ideal gas law assumes no intermolecular forces, but as pressure increases (and volume decreases), these forces become more significant. The energy added to the system during compression isn’t solely used to increase translational kinetic energy (temperature); some is used in overcoming these attractive forces.

So, while the simple scaling might suggest a 12x increase in temperature when volume is reduced by a factor of 12, the actual physics involving work done, energy distribution among molecular degrees of freedom, and non-ideal gas behavior result in a less pronounced temperature increase.

ans 2

When a gas is compressed adiabatically, the work done on the gas increases its internal energy, which is reflected as an increase in temperature. The temperature increase is not directly proportional to the compression ratio because of the nature of the adiabatic process, which is governed by the specific heat capacities and the adiabatic index (\(\gamma\)).

1. Internal Energy and Temperature: The internal energy of a gas is predominantly kinetic energy at the molecular level. When the gas is compressed, the molecules are forced closer together, increasing their kinetic energy, which is observed as an increase in temperature.

2. Repulsive Forces: As the molecules are compressed, they come closer together, and the repulsive forces (mainly Pauli repulsion due to electron cloud overlap) become significant. These repulsive forces are what the work is primarily overcoming during compression. The increase in repulsive force with decreasing volume is a significant factor in the non-linear relationship between temperature and volume during adiabatic compression.

3. Attractive Forces: While there are attractive forces (like van der Waals forces) between molecules, these are generally more significant in liquids and solids than in gases. In the context of gases being compressed, the dominant interactions as the molecules are forced closer are repulsive.

4. Energy Distribution: Not all the work done on the gas goes into translational kinetic energy (which would directly increase temperature). Some energy also goes into rotational and vibrational modes, especially in polyatomic gases, which partially explains why the temperature increase isn’t proportional to the compression ratio.

In summary, during adiabatic compression, the primary interaction at the molecular level is the increase in repulsive forces as molecules are forced closer, requiring work to be done against these forces. This work translates into an increase in internal energy, observed as an increase in temperature, but the relationship is governed by the specific heat capacities and is not a simple direct proportion.

Plasma

Yes, the concept of energy being diverted into changing electron orbits is highly relevant to plasmas. Plasma, often referred to as the fourth state of matter, consists of a mixture of free electrons, ions, and neutral atoms or molecules, where a significant fraction of the particles are ionized.

In plasmas:

1. Electronic Excitation: Electrons can gain energy through collisions or electromagnetic fields, moving to higher energy orbits and then falling back to lower levels, emitting photons in the process. This is a common mechanism for light emission in plasmas.

2. Ionization: Energy can be absorbed by atoms or molecules to overcome the ionization energy, resulting in the release of electrons and the creation of ions. This is a key feature distinguishing plasmas from other states of matter.

3. Recombination: Free electrons can recombine with ions, releasing energy in the process, often in the form of light.

4. Thermal Energy: Like gases, plasmas have translational, rotational, and vibrational modes of energy, but the presence of significant ionization and electronic excitation makes their behavior more complex.

The balance of energy between these various processes in plasmas is crucial for understanding their properties and behavior, particularly in fields like astrophysics, fusion research, and plasma technology.

References

• Introduction to Quantum Mechanics by David J. Griffiths While not solely focused on gases, this textbook provides a clear introduction to quantum mechanics, which is fundamental for understanding the quantum theory of gases.

• Statistical Mechanics by Kerson Huang: This book offers an in-depth look at statistical mechanics, including the quantum theory of gases, with clear explanations and applications.

• Thermodynamics and an Introduction to Thermostatistics by Herbert Callen (2nd Edition): A classic text providing a rigorous foundation in thermodynamics and its connection to statistical mechanics, the basis for understanding the behavior of gases.

• Introduction to Thermal Physics by Daniel Schroeder: A clear and engaging introduction to thermodynamics and statistical mechanics, with a focus on conceptual understanding. Great for beginners

• Quantum Gases: Finite Temperature and Non-Equilibrium Dynamics by Nick Proukakis et al.: A comprehensive resource delving into the complexities of quantum gases. This book covers advanced topics like Bose-Einstein condensates and non-equilibrium phenomena.

• Statistical Mechanics by R.K. Pathria and Paul Beale: An authoritative text covering statistical mechanics in great depth, offering the necessary groundwork for understanding the quantum behavior of gases.

• Quantum Theory of Many-Particle Systems by Alexander Fetter and John Walecka: While not focused solely on gases, this book provides a deep exploration of quantum mechanics applied to systems with many particles, laying important theoretical foundations for understanding quantum gases.