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Ideal Gases

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Ideal Gases

From a thermodynamic point of view, gases are systems characterised by the randomness of the movement of particles, that lose almost no energy while interacting among themselves. This allows us to study their statistical features in a simple manner, and extract uninvolved mathematical properties.

There were huge advances in experimental thermodynamics between the 17th and 19th centuries. These led scientists to discover important laws regarding the relationships among quantities that describe gases. In particular, they discovered that under certain conditions, there were laws that related the pressure, volume, and temperature of a gas. These laws are approximate and can be derived from a theoretical model called the 'Ideal gas model'.

The ideal gas model concerns macroscopic statistical quantities like temperature, but it is derived from a microscopic model for all the particles that constitute gases, better known as 'kinetic molecular theory'. Understanding this connection between the microscopic and the macroscopic is key to understanding the ideal gas laws.

What is kinetic molecular theory?

The kinetic molecular theory is the theory that studies matter by considering it as a many-particle system. The adjective kinetic makes reference to the fact that many of the features of the system that are to be studied are extracted from the state of movement of the microscopic particles.

Main quantities

The main quantities that are used in kinetic molecular theory:

  • Position: Although it is not used for determining any of the main thermodynamical properties like temperature or pressure, it can be used as a first measure of determining whether a system is in a solid, liquid, or gas state.
  • Speed: The speed of particles is one of the main microscopic quantities kinetic molecular theory is concerned with. The distribution of speeds of particles throughout a sample will help to determine the pressure and its average value will help determine the temperature.
  • Kinetic energy: The kinetic energy of particles in matter comes in many forms, and they are relevant in different ways for different phases. For instance, the kinetic energy of the particles of a solid is restricted because they are highly localised in space, and do not move linearly. Nevertheless, their particles vibrate. This is a big contribution to the total kinetic energy.
  • Interaction/potential energy: The energy of interaction, or potential energy, is the energy that captures how particles in a sample of matter interact among themselves. While in gases this energy to not be very important, in solids it is bound to be the main source of energy in the system; the strength of the interaction is what keeps the particles close to each other, and still.

Macroscopic interpretation

Now that we've seen the main microscopical quantities the theory is concerned with, let's connect them with the macroscopical quantities we study in thermodynamics. Among others, these are:

  • Temperature: A measure of the average kinetic energy of the particles that constitute a system. In gases, it measures how quickly the particles will move around (and get away from each other if they are not bound within a certain volume). In solids, the higher the temperature, the higher the vibrational energy. This is a first approximation to the changes of phase. If a solid reaches a certain temperature, the vibrational energy will be high enough to allow the particles to break the structure and turn into a liquid phase.
  • Pressure: It is a measure of the average force per unit of area that the particles exert on the boundaries of the space they occupy. It is related to the distribution of the kinetic energy of the different particles throughout the sample, and it informs the distribution of this energy. Although it is defined as a measure of the exerted force per unit of area (which makes it useful for engineering applications) it can also be understood as a measure of the density of kinetic energy.
  • Volume: It is a measure of the space the particles of a system occupy. The distribution of the position of the particles gives all the information needed to specify the volume. The notion of volume is also closely related to the spacing between particles in a sample, which influences the strength of the interactions and varies from one phase to another (spaces between particles of a sample in a solid phase are fixed and small, while for a gas phase they are random and larger on average).
  • Total energy: A global measure that takes into account the kinetic energy, the potential energy that captures the nature of the interactions among particles, and the distribution of these quantities for every particle in the sample.

What is the relevance of ideal gases?

The model of ideal gases is a very good approximation to the behaviour of real gases, as it is derived from simple assumptions for the (associated) microscopic kinetic molecular theory. Understanding this model allows us to grasp the relevance of the thermodynamical quantities and their relationships. It is also crucial to understand why (and under which conditions) the approximation fails, and a more complex model is needed.

Assumptions of the model

The kinetic molecular theory model for ideal gases assumes the following:

  • There are no intermolecular forces, i.e., forces acting among the molecules/particles that form the system.
  • There are collisions among the particles, but their duration is small compared with respect to the time between collisions.
  • The collisions among the particles are elastic, which means that there is no loss of energy involved.
  • The motion of the particles is random and is subject to Newton's laws.
  • The particles do not have a volume; they are point-like (infinitesimally small).

These assumptions translate into different consequences at a macroscopic level.

  1. It is convenient to consider averages for defining the macroscopical quantities because the movement is random.

  2. The applicability of the laws of Newton allows us to study transfers of momentum among particles and between particles and their boundary.

  3. The elasticity of collisions implies that the kinetic energy is conserved (Newton's third law). For ideal gases, this is equal to the total energy because there is no potential energy (no intermolecular forces).

Law of ideal gases

The law of ideal gases is an equation that captures the relationship between the three main macroscopical quantities: pressure, temperature, and volume. It also includes the particle content of the system under study through the use of the number of mols, or n. The equation is shown below.

where P is the pressure of the gas, V is the volume, T is the temperature (in Kelvin) and R is the ideal gas constant (with an approximate value of 8.31 J / mol·K ).

This equation can be derived for n moles of particles that behave according to the assumptions listed before, and by performing statistical analysis to extract thermodynamical quantities. From this equation, leaving the particle content and one of the thermodynamical quantities fixed, one can arrive at the three laws of ideal gases:

  • Boyle's law (temperature is fixed).
  • Charles' law (pressure is fixed).
  • Gay-Lusaac's law (volume is fixed).

Applicability

Gases seem to obey the laws of ideal gases for experimental conditions in a certain regime. The experimental laws that led to the deduction of the complete law of ideal gases were studied in conditions of temperature, pressure, and volume in this type of regime.

However, generally speaking, the ideal gas approximation fails for two main reasons:

  1. The particles that form gases are not point-like; they have a volume.
  2. There are intermolecular forces among the particles.

These two contributions were included in the Van der Waals equation parametrised by the factors a and b:

This equation yields the ideal gas equations upon setting a=0 and b=0. It can be proven that for a situation where the molar volume (volume occupied by 1 mol of substance) is much larger than a and b, they can be effectively considered as zero. So, we can use the ideal gas law. The situation where 1 mol of a substance occupies a large volume does match our assumptions of the model because:

  1. Even if the particles do have a volume and are not point-like, their volumes are negligible compared with the large volume available to the substance.
  2. Even if there are intermolecular forces among the particles, the large volume available allows the particles to be apart from each other, so as to render the value of intermolecular forces as approximately zero.

Ideal Gases diagram showing the difference between Ideal Gases and real gases StudySmarter

Ideal gases and real gases. [energyeducation.ca]

Ideal Gases - Key takeaways

  • The ideal gas model is a theoretic mathematical model that describes the relationship among macroscopical quantities for gases that obey certain simplifying assumptions.
  • The kinetic molecular theory is the theoretical framework that allows us to systematically study the particles that form a many-particle system, and to extract thermodynamical quantities with statistical techniques.
  • The assumptions of the kinetic molecular theory for ideal gases consider that the particles are infinitesimally small and do not interact among themselves (they have no potential energy). The kinetic energy is not modified because the collisions that happen within the boundary of the occupied volume are elastic.
  • The Van der Waals equation captures how gases deviate from their ideal behaviour through some parameters that quantify the intensity of the intermolecular forces, and the real size of particles.

Frequently Asked Questions about Ideal Gases

Real gases are not made of point-like particles, and there are intermolecular forces present.

No, they can be described by this law under certain thermodynamic conditions that match the assumptions of the ideal gas model.

They do not exist as universal systems, but they describe the behaviour of gases within a certain regime of thermodynamic conditions.

Final Ideal Gases Quiz

Question

Choose the correct statement.

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Answer

Kinetic molecular theory aims to extract macroscopical properties from the statistical study of microscopical properties.

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Kinetic molecular theory has to do with speeds, individual energies, and positions.

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Temperature measures the average kinetic energy of particles.

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Answer

The ideal gas model assumes that the particles have an infinitesimally small volume.

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Choose the correct statement.

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Answer

When the molar volume is large, gases can be described as ideal.

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Question

What are the experimental laws that describe processes of ideal gases?

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Answer

Boyle's law, Charle's law, and Gay-Lusaac's law.

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What is the usual description of pressure?

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Answer

The average force per unit of area exerted on the boundaries of the occupied volume.

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Question

Can one apply Newton's laws to ideal gases?

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Answer

Yes, it's one of the main assumptions of the model.

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Question

What is the value of the constant of ideal gases?

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Answer

8.31 J / K·mol

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What are the two main reasons why real gases deviate from ideal gases?

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Answer

There are intermolecular forces among particles, and the particles occupy a non-negligible volume.

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Question

Why is it convenient to use statistical quantities in the analysis of gases?

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Answer

Because the movements of particles are random.

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Question

Name the energy particles possess by virtue of their movement

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Answer

Kinetic energy.

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What is the name of the energy that captures the interactions among particles?

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Answer

Potential energy.

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What is the potential energy of ideal gases?

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Answer

Zero, because there are no intermolecular forces.

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Question

What is the name of the equation of gases that captures the behaviour of real gases?

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Answer

Van der Waals equation.

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