Physics 13: 6-13 The Ideal Gas Law

Introduction

Le puissance motrice de la chaleur est indépendente des agens mis en oeuvre pour le réaliser: sa quantité est fixée uniquement par les temperatures entre lesquels se fait en dernier résultat le transport de calorique.

The horse-power equivalent of heat is independent of the agents used to carry out the work done; its quantity is fixed only by the temperatures of the initial and final result of the heat transfer.

— Sadi Carnot, Reflections on the motive power of fire, 1824

Kinetic Energy and Gas Behavior

So far, we have considered ideal situations: pure gases, behaving like tiny billiard balls. Now we are going look more closely at the real situation, and take into account some of the adjustments we need to make to the Ideal Gas Law when we are dealing with mixtures of gases, molecules which aren't point masses, and which may have electrical charges that affect their interaction.

Mixtures of Gases: Partial Pressures

John Dalton, the "father of modern chemistry", was able to observe that when two gases are mixed in the same container (and therefore at the same temperature and volume), the total pressure is equal to the sum of the individual pressures exerted by each gas separately. It is as though neither gas interacts with the other at all.

P_{tot} = P_{1} + P_{2}

P_{tot} = \frac{n_{1} RT}{V} + \frac{n_{2} RT}{V}

Any pressure differences are dependent on a difference in amounts of the gases.

Many gases are prepared by techniques which release them in aqueous solutions, so that water vapor is present as well as the gas being prepared. There is a relationship between the amount of vapor above any liquid (not just water) and the temperature. In a lab situation, you would look up vapor pressure in a reference book; for example (from my chemistry text) the vapor pressure (in mm Hg) of water is given in varying increments for the range of temperatures between 0 and 110 °C.

Gas produced by the reaction in one erlenmeyer flask has been collected in an inverted flask. As the collection continues, the gas displaced the water in the inverted flask. We can measure the amount of water displaced and determine the volume of the gas. When the water level stopes moving, the reaction is over, and the pressure inside the inverted flask is equal to the pressure outside the flask. By measuring the current barometric pressure, we can determine the pressure inside the flask. With pressure and volume and a lab measurement for temperature, we can determine the moles of gas produced, but we have to take into account the amount of pressure exerted by the water vapor trapped inside the inverted flask, along with the generated gas.

For example, 225 mL of hydrogen is collected over water at 15 degrees C, and 740 mmHg. We need to know the number of moles hydrogen produced by this (unspecified) reaction.

First, realize that the pressure in the collection flask is due both to H₂ and to H₂O vapor. At 15 deg C, water vapor pressure is (from my handy appendix) 13 mmHg. So only 740-13 mmHg = 727 mmHg is due to the H₂ gas.

Now calculate the number of moles from the ideal gas law. Since we are using R in liters, atmospheres, moles, and Kelvin, we must convert each value to the proper units: $n = \frac{(727 mmHg * \frac{1 atm}{760 mmHg}) (225 mL * \frac{1 L}{1000 mL})}{(0.821 \frac{L * atm}{moles * K}) (273 K + 15 K)} = 0.00910 moles$

As mentioned above, the partial pressure is a direct indication of the amount of gas present. If you know PV and RT, you know n; conversely, if you know the mole fraction, you can get any of the remaining three if you know the other two factors (R being constant).

Kinetic theory of gases

Gas molecules are like little billiard balls which are constantly in motion. Since the motion is random, we cannot predict which direction a specific molecule will move. Each molecule's kinetic (movement) energy is E = mv²/2. Since this motion is dependent on temperature, E is dependent on temperature as well: E = cT, where c is a constant. We can get the dependency of mass, velocity and temperature by equating the two formulae for energy: $E = \frac{m v^{2}}{2} = cT$ Thus (average) velocity in the gas is directly dependent on temperature: $v = \sqrt{\frac{2 cT}{m}}$

How can we determine c?

Analysis of the three-dimensional motion of a molecule averaged over billions of gas molecules striking the wall of the contain leads to the relationship between force , pressure, and volume: $F = Nm * \frac{v^{2}}{3 l}$ where v is the average velocity, m is the mass of each molecule, N is the number of molecules, and l is the length of the side of the cube of volume containing the N molecules. [The derivation of this relationship is given in your text.]

Since P = F/A, we can write $P = Nm * \frac{v^{2}}{3 lA} = \frac{Nm v^{2}}{3 V} \Rightarrow PV = \frac{Nm v^{2}}{3}$

Now we do some equivalency analysis, substituting in "equivalent" values from the ideal gas law and the definition of kinetic energy: $\begin{array}{l} PV = NkT \\ KE = \frac{m v^{2}}{2} \end{array}$

We can replace PV with nKT and mv² with 2 KE: $Nkt = \frac{N * 2KE}{3}$

The number of molecules factors out. Solving for KE $\frac{3kT}{2} = KE$

This allows us to easily equate temperature with the average kinetic energy of molecules in the gas.

Diffusion

We can use the same technique as before in comparing the same gas under different conditions, or different gases under similar conditions. Isolate the conditions that are changing from those that are not, and equate the two situations. For example, if we have a sample of gas (m not changing) at two different conditions of temperature, we can determine the ratio of their velocities:

\begin{array}{l} \frac{v_{1}}{\sqrt{T_{1}}} = \sqrt{\frac{2 c}{m}} = \frac{v_{2}}{\sqrt{T_{2}}} \\ \frac{v_{1}}{v_{2}} = \frac{\sqrt{T_{2}}}{\sqrt{T_{1}}} = \sqrt{\frac{T_{2}}{T_{1}}} \end{array}

When a gas escapes from its container (effuses) through a small hole, the rate of flow depends on the pressure of the gas (a function of the amount of gas and the temperature) and the relative speeds of the particles (which is dependent on temperature and mass). Since mass is related to molar mass, knowing the rate of effusion can help us determine the molar mass:

\frac{m_{1} {v_{1}}^{2}}{2} = \frac{m_{2} {v_{2}}^{2}}{2} \Rightarrow m_{1} {v_{1}}^{2} = m_{2} {v_{2}}^{2}

Again, solving for ratios of the same type,

\frac{{v_{1}}^{2}}{{v_{2}}^{2}} = \frac{m_{2}}{m_{1}} \Rightarrow \frac{v_{1}}{v_{2}} = \sqrt{\frac{m_{2}}{m_{1}}}

The rates are directly dependent on the molar masses.

This gets us the average velocities for our gas samples. Obviously, not all molecules in a sample travel at the same speed. A perfectly statistical distribution looks a like bell curve--it's symmetrical around the high point or average speed. If you closely at the Maxwell distribution used for gases (Figure 13-12), you'll notice that it is NOT symmetrical around this point or any particular point. A commonly used form for the distribution is n = Ne^-E/RT , where n is the number of molecules with energy in excess of some baseline energy E, T is the absolute temperature (in Kelvin), N is the total number of molecules involved, and e is the exponential constant 2.71828.....; never mind why for now.

Real Gases

Real gases, of course, vary in their molar volumes (we have been assuming the volume is zero, and obviously it cannot be) and have forces of attraction (affinity) which alter their interactions with each other, so that they don't behave quite like neutrally charged billiard balls. The general result is that the volumes actually observed are slightly less than those predicted by the Ideal Gas Law.

Attractive forces pull the molecules together and slow down their interactions, so the molecules travel shorter distances than we would otherwise expect. This lowers the volume required by the gas. On the other hand, since the molecules actually do take up space, their real positive volume increases the volumen required by the gas. These two opposing forces tend to make the real gas more like the ideal model, except at extreme pressure or temperature, where one or the other factor becomes dominant.

In order to account for these deviations, we can use the van der Waals equation $[P + a {(\frac{n}{V})}^{2}] [\frac{V}{n} - b) = RT$ where a and b are constants, a/V is the attractive force between molecules and b is the additional volume due to the real volume of each gas molecule.

Practice with the Concepts

Discussion Points

Is it possible to boil water at room temperature (20 °C) without heating it?
The escape velocity of the Moon is about 1/5 that of Earth. How does this explain the fact that the Moon has no atmosphere?

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Physics

Chapter 13: 6-13