Kinetic Theory of an Ideal Gas

According to kinetic theory, gas pressure is the result of the bombardment of the container walls by constantly moving molecules. 

Waterston s paper, with the reviewers comments, was discovered in the Royal Society s files by Lord Rayleigh in 1892. (Rayleigh codiscovered argon and received the 1904 Nobel Prize in physics.) 

The statements that there are no intermolecular forces and that the volume of molecules is negligible are simplifications that lead to the ideal gas law. But intermolecular forces are needed to explain how we get the liquid state from the gaseous state it is intermolecular forces that hold molecules together in the liquid state. 

According to Newton, the pressure of a gas was due to the mutual repulsions of the gas particles (molecules). These repulsions pushed the molecules against the walls of the gas container, much as coiled springs packed tightly in a box would push against the walls of the box. This interpretation continued to be the dominant view of gas pressure until the mid-nineteenth century. 

Despite the dominance of Newton s view, some people followed the kinetic interpretation. In 1738, Daniel Bernoulli, a Swiss mathematician and physicist, gave a quantitative explanation of Boyle s law using the kinetic interpretation. He even suggested that molecules move faster at high temperatures, in order to explain Amontons s experiments on the temperature dependence of gas volume and pressure. However, Bernoulli s paper attracted little notice. A similar kinetic interpretation of gases was submitted for publication to the Royal Society of London in 1848 by John James Waterston. His paper was rejected as nothing but nonsense.  

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The kinetic theory of an ideal gas leads directly to an expression for the so-called root-mean-square velocity v of the gas molecules, viz. 

True adsorption is a "mass action" process rather than a mass transfer process. What this means is that it will occur even in the absence of a concentration gradient between the bulk gas and the surface. It comes about due to the rapid and chaotic motion of the fluid phase molecules, and their impingement on the surface. From the elementary kinetic theory of an ideal gas we can compute the number of molecules impinging upon a surface per unit time per unit area at a given temperature and pressure. It is ... 

The kinetic theory of an ideal gas (a gas that follows the ideal gas law) is based on five postulates. 

The radius of a typical molecule of a gas is 2.00 A. (a) Find the volume of a molecule assuming it to be spherical. For a sphere, V = 4/3 irr. (b) Calculate the volume actually occupied by 1.00 mol of these molecules, (c) If 1.0 mol of this gas occupies 22.4 L, find the fraction of the volume acmally occupied by the molecules, (d) Comment on your answer to part (c) in view of the first statement summarizing the kinetic-molecular theory of an ideal gas. 

According to the kinetic-molecular theory of gases, the internal energy of an ideal gas, U,... 

The kinetic molecular theory (KMT see Sidebar 2.7) of Bernoulli, Maxwell, and others provides deep insight into the molecular origin of thermodynamic gas properties. From the KMT viewpoint, pressure P arises merely from the innumerable molecular collisions with the walls of a container, whereas temperature T is proportional to the average kinetic energy of random molecular motions in the container of volume V. KMT starts from an ultrasimplified picture of each molecule as a mathematical point particle (i.e., with no volume ) with mass m and average velocity v, but no potential energy of interaction with other particles. From this purely kinetic picture of chaotic molecular motions and wall collisions, one deduces that the PVT relationships must be those of an ideal gas, (2.2). Hence, the inaccuracies of the ideal gas approximation can be attributed to the unrealistically oversimplified noninteracting point mass picture of molecules that underlies the KMT description. 

As a starting point in the study of equations of state of real gases, we will consider a hypothetical gas known as an ideal gas. We will develop the equation of state of an ideal gas in two ways, first from experimental evidence and then from kinetic theory. The form of the equation for ideal gases will be used as the basis of equations for real gases. 

The previous equations which describe the behavior of an ideal gas now will be verified using the kinetic theory of gases. This will illustrate the reasons for the previously given three conditions imposed on the molecules of an ideal gas. Also, you will gain an understanding of the meanings of pressure and temperature. 

Thus, the equation of state of an ideal gas can be derived from kinetic theory. All three properties of an ideal gas were used in the development of the equation. 

We have shown from kinetic theory that the ideal gas equation has the correct form. The behavior of most real gases does not deviate drastically from the behavior predicted by this equation. So the best way of writing an equation of state for a real gas is to insert a correction factor into the ideal gas equation.1 This results in... 

Boyle s Equation —Charles Equation—Avogadro s Law — The Equation of State for an Ideal Gas — Density of an Ideal Gas — Kinetic Theory of Gases Mixtures of Ideal Gases 100... 

Before ending this discussion of gases, it s worthwhile expanding on a point made earlier The behavior of a real gas is often a bit different from that of an ideal gas. For instance, kinetic-molecular theory assumes that the volume of the gas particles themselves is negligible compared with the total gas volume. The assumption is valid at STP, where the volume taken up by molecules of a typical gas is only about 0.05% of the total volume, but the assumption is not valid at 500 atm and 0°C, where the volume of the molecules is about 20% of the total volume (Figure 9.14). As a result, the volume of a real gas at high pressure is larger than predicted by the ideal gas law. 

Example 2.1. To estimate the number of gas molecules hitting the liquid surface per second, we recall the kinetic theory of ideal gases. In textbooks of physical chemistry the rate of effusion of an ideal gas through a small hole is given by [12]... 

The temperature of an ideal gas is proportional to the average kinetic energy of the particles in a sample of the gas. The relationship between kinetic energy and temperature is derived from the Kinetic Molecular Theory of Gases and is given by ... 

The kinetic molecular theory describes a hypothetical gas called an ideal gas. In an ideal gas, the gas particles take up hardly any space. Also, the particles of an ideal gas do not attract each other. 

From the kinetic theory of gases we know that the average translational kinetic energy of a molecule of an ideal gas is given... 

An example of this type of model is the kinetic molecular theory, a simple model that attempts to explain the properties of an ideal gas. This model is based on speculations about the behavior of the individual gas particles (atoms or molecules). The postulates of the kinetic molecular theory can be stated as follows ... 

Know the characteristics of an ideal gas and the rules that govern its behaviour (kinetic theory of gases). 

The pair of molar heat capacities Cy and Cp for an ideal monatomic gas can be calculated from the results of the kinetic theory of gases and the ideal gas equation of state. From Section 9.5, the average translational kinetic energy of n moles of an ideal gas is... 

To better understand the complex behavior of gases, scientists have theorized a model of an ideal gas. This model is called the kinetic molecular theory. In the kinetic molecular theory, an ideal gas lacks certain real gas characteristics. Ideal gas has the following four characteristics not shared by a real gas ... 

As early as 1738, Daniel Bernoulli (1700-1782) envisioned gaseous molecules in ceaseless motion striking the walls of their container and thereby exerting pressure. In 1857, Rudolf Clausius (1822-1888) published a theory that attempted to explain various experimental observations that had been summarized by Boyle s, Dalton s, Charles s, and Avogadro s laws. The basic assumptions of the kinetic-molecular theory for an ideal gas follow. 

From kinetic theory, the pressure of an ideal gas can have the form... 

From the kinetic theory of gases, without a knowledge of the law of distribution of velocities (i.e. of the way in which the number depends on v), we have found that the product of pressure and volume is a function only of the mean kinetic energy of the gas. But we have also an empirical law, the law of Boyle (1660) and Mariotte (1676), viz. at constant temperature the product of the pressure and volume of an ideal gas is constant. We must conclude from this that U, the mean kinetic energy per mole, depends only on the temperature of the gas. 

The kinetic molecular theory describes an ideal gas in which gas particles exhibit no interactive or repulsive forces and the volumes of the individual gas particles are assumed to be negligible. 

The viscosity of an ideal gas by kinetic molecular theory is given by... 

The kinetic energy of one mole of an ideal gas at temperature T (K) is givmi by its translational energy, which according to the kinetic theory of gases is... 

This agreement between experiment and theory supports the validity of the assumptions made in the kinetic molecular model about the behavior of gas particles, at least for the limiting case of an ideal gas. 

The equation contains two corrections bis a. correction for the non-negligible size of the molecules alV corrects for the fact that there are attractive forces between the molecules, thus slightly reducing the pressure from that of an ideal gas. The equation is named for the Dutch physicist Johannes Diderik van der Waals (1837-1923). See also gas laws kinetic theory. 

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