Gases Ideales

The Ideal Gas Law and the Kinetic Theory of Gases

We continue our discussion of temperature and heat but we are going to approach it in a different way. From last week, we looked at the global or bulk properties of materials when I raise or lower their temperature. Now we are going to discuss the affects of raising the temperature on a molecular level.

Most of the quantities that we will look at are not derived quantities but empirical quantities. That is, they are laws that come about from experiments and not theory. So, for the most part, the laws that we use are valid, except in some wild extremes.

First, to discuss what happens to the molecules in a gas, we need a way to quantify the number of molecules in a certain amount of gas. Therefore, we now define the mole.

To begin, the relative masses of the atoms of different elements can be expressed in terms of their atomic masses, which indicates the mass of one atom as compared to another. The unit on this scale is called the atomic mass unit. The molecular mass of a molecule is the sum of the atomic masses of its atoms.

A mole, or one mole of a substance, contains as many particles as there are atoms in 12 grams of carbon-12. Experiment show that there are 6.022x1023 atoms in 12 g of carbon-12. Therefore, one mole of hydrogen contains 6.022x1023 atoms of hydrogen. One mole of water contains 6.022x1023 molecules of water…and so on. Furthermore, one mole of a substance has a mass in grams that is equal to the atomic of molecular mass of the substance.

Example, how many grams are there in one mole of water?

The Ideal Gas Law

The ideal gas law expresses the relationship between the absolute pressure, the Kelvin temperature, the volume, and the number of moles in a gas.

First, when we were discussing the Kelvin scale, we saw that is was an extrapolation of what would happen to the pressure in a gas, if the temperature kept decreasing. We saw that the absolute pressure, P, is directly proportional to the Kelvin temperature, T, for a fixed volume of gas. (P T)

Also, we can infer what would happen to the pressure inside a gas if we increased the number of molecules, or moles. The absolute pressure of an ideal gas is proportional to the number of molecules, or the number of moles, n, of the gas. (P n)

Finally, it is possible to increase the pressure in a gas if I reduce the volume, if the temperature and number of molecules stay constant. So the absolute pressure in an ideal gas is inversely proportional to its volume. (P 1/V).

We can express these as a single proportionality; writing it down:

Also, the constant of proportionality is called R, the universal gas constant.

If we replace the number of moles with the number of particles, N, we can rewrite the ideal gas law as:

Where the constant, R/NA, is called Boltzmann’s constant and has a value of 1.38x10-23 J/K, and is represented by the symbol, k. So the ideal gas law becomes:

Origin of the Ideal Gas Law

The work of several people led to the formulation of the ideal gas law. The scientist Robert Boyle discovered that at a constant temperature, the absolute pressure of a gas is inversely proportional to the volume. Boyle’s Law states:

P1V1 = P2V2

The curve that passes through the initial and final points is called an isotherm (the gas expands slowly enough to allow the system to remain in thermal equibrium). Because it will take a certain amount of work to expand or contract the system, the work done by the gas as its volume changes is given by the integral:

Furthermore, there are several different paths you can take to get from the initial to the final states. The work done by a system depends on the initial and final states and on the path followed by the system to these states. Similarly, we can also account for the amount of heat added or subtracted to the system as well. The energy transferred by heat also depends on the initial, final and intermediate states of the system. So we now have two ways energy is transferred from the system to the surroundings:

1. Work is done by the system on its surroundings

2. The system transfers heat to the surroundings.

This will lead us the first law of thermodynamics a little later.

Another scientist, Jacques Charles, discovered the relationship between the pressure and temperature in an ideal gas. This Charles’ Law states:

V1/T1 = V2/T2

The Kinetic Theory of Gases

At any time in a container, the molecules are moving at some speed for a given temperature. The physicist James Clerk Maxwell was the first to find the distribution of speeds within a large collection of molecules at a constant temperature. In order to develop a model for the motion of the particles in a gas, we have to define a few assumptions:

1. The volume of the molecules is negligible compared to the volume of the container.

2. The molecules obey Newton’s Laws…something

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