Pressure of several gases in the same container


You have probably studied the behavior of a single ideal gas in a container, how its pressure, temperature and volume change when one or another is manipulated.

For example, Charles law, P1V1 = P2V2, tells us that the pressure of a system will increase as the volume is decreased. That is, if V2 < V1, then P2 > P1.

But what about mixtures of gases? For example, how do we characterize the pressure of a mixture of argon (Ar) and helium (He), and can they be treated separately?

Dalton's law of partial pressures, due to John Dalton (1766 - 1844), says that the total pressure of a system of non-interacting gases is just the sum of the partial pressures of each of the components.

The partial pressure of a gas is just its pressure calculated under the assumption that it's the only thing in the container.

Partial pressures in the atmosphere


A nice example is the atmosphere of Earth (although the "container" is a little bit hard to define. Near the surface of our planet, the air is about 78% nitrogen (N2), 21% oxygen (O2), 1% argon (Ar), and less than a percent other gases such as carbon dioxide (CO2). The total pressure near sea level is 1 atm., therefore, the law of partial pressure says that 78% of that (0.78 atm) is due to N2 molecules, 21% is due to O2 molecules, and 1% is due to argon and the rest of the gases.

Dalton's law of partial pressures

The total pressure of a mixture of non-interacting ideal gases A, B, C, ... is the sum of the partial pressures of each gas:

Ptotal = PA + PB + PC + ...

where

and so on ...

The basic assumption behind Dalton's law of partial pressures is that ideal gas particles are so far apart, except at extremely-high pressures, that collisions between them are not as important as collisions with the walls of the container. We treat them as if they are completely independent.

Example 1

11.0 g of N2 and 5.0 g of O2 are let into an evacuated container of volume V = 12.5 liters. Calculate the partial pressures of each gas and the total pressure in the container if the temperature is maintained at T = 300K.


Solution: Aiming for calculating each pressure with the ideal gas law, PV = nRT, first we'll need to calculate the number of moles of each gas. The number of moles of N2 is:

and O2:

Now we use the ideal gas equation to calculate the pressures:

Using R = 0.0821 L·atm·mol-1·K-1 for the gas constant (often much more convenient than the SI-units version), and the temperature and volume given, the pressure of N2 is

Both results are

The total pressure is just the sum of those partial pressures:

Example 2

The pressure relief valve on a cylinder fitted with a piston activates at a pressure of 7.5 atm. If the initial pressure and volume of the cylinder are P1 = 1.2 atm and V1 = 10 liters, to what volume can the gas be compressed before the relief valve (a safety device to prevent over-pressurization of the cylinder) activates?

We often engineer gas-handling systems with pressure-relief valves to protect from over-pressurization, which could cause failure of the system and could hurt someone. Here's the sketch of what's going on:

The

We're looking for V2, so we can rearrange Boyle's law to get it (I like to rearrange formulas before plugging in numbers – then I know I'm on the right track):

Now plugging in what we know gives

So the volume can be reduced to

before the relief valve opens.


Charles' law


Charles' law says that, if we keep the number of particles and the pressure constant, then the ratio of volume (V) to temperature (T) remains constant:

If we hold the number of particles (n moles or N particles) and the pressure (P) constant, then comparing state 1 of a gas system with state 2 gives us Charles' law:

If we rearrange the first expression to V = ct, where c is some constant of proportionality, and plot a graph of V vs. T, we see that the relationship is linear.

It makes sense, if volume changes, the temperature must change proportionally to keep the ratio constant.

Some examples of how to use Charles' law are given below.


Example 3

How much (in percent) does the volume of a fixed amount of gas, held at a constant pressure, expand when the temperature is raised from 30˚C to 37˚C


Here's the situation. It might seem like we don't have enough information, but we do. We don't need the initial volume because we're only trying to find how much any volume of gas would expand (at constant pressure) if the temperature is raised from 30˚ to 37˚C.

We rearrange Charles' law to find the final volume:

Now plug in the temperatures, letting x be the initial volume. The Celsius temperatures are fine because we're calculating a ratio of temperatures, and because the size of the Celsius degree is the same as the Kelvin.

The ratio of temperatures gives us our result.

So for an initial volume of 1 liter, the final volume would be 1.233 liters, for a 23.3% change in volume.

Example 4

If a gas is expanded to twice its original volume while keeping the pressure constant, by how much, and in which direction (warmer / cooler) must the temperature change?


Here is the picture, gas expanding in a cylinder with a movable piston:

In this case it seems like we have even less information to go on, but it's still OK. Take a look.

Charles' law can be rearranged to give the final temperature like this:

Then we can plug in the unknown volumes, but their ratio has to be ½ – that's all that's needed.

So the final temperature will be half of the initial temperature, whatever that was.

The cooling of gases when they expand is the basic idea behind refrigeration and air conditioning.



Gay-Lussac's law


The Gay-Lussac law of gases says that if we hold volume and number of particles (or moles) of gas constant, then the ratio of pressure to temperature will be constant:

For a gas in state 1, with P1 and T1, we can make this statement, analogous to Charles' law:

We can rationalize the Gay-Lussac law by considering what happens to gas atoms or molecules when we raise the temperature. As T increases, the average velocity of particles increases, therefore collisions with the container walls exert more force on the walls, and the collisions are more frequent. That all adds up to higher pressure.

As for Charles' law, this relationship is linear. We would expect a graph of P vs. T to be linear, with a slope equal to the proportionality constant.

Example 5

Calculate the pressure drop we would expect to see if a quantity of a gas is is cooled from 42˚C to 20˚C at constant volume.


Here's the picture for this problem. Remember that we're forcing volume to remain constant, so we're not letting the piston move in this case.

The Gay-Lussac law can be rearranged to solve for the final pressure, P2, like this:

We don't need to know the initial pressure, just that the equation shows that it will be reduced (in this case) by the ratio of the final and initial temperatures:

The result is:

... so this reduction of temperature would result in the pressure dropping to about 48% of its original value.


Avogadro's law


Finally, Avogadro showed that if we hold pressure and temperature constant, the ratio of volume to number of moles of gas (or number of particles, N) is constant:

Avogadro actually used this observation to develop the concept of the mole in chemistry.

Just like we did for the previous gas laws, we can rewrite Avogadro's law as an equation involving initial and final states of a gas – in this case as gas is added or removed from the container

Examples of using Avogadro's law are very similar to the previous examples involving ratios.


The ideal gas law


Finally, we can put all of these observations together to develop the ideal gas law. It says that the pressure-volume product, PV, is proportional to the product of the number of particles (or moles) and the temperature. The constant of proportionality really just helps us get the units right.

The ideal gas law can be expressed in two main ways, in terms of particles

,

where N is the number of particles, and in terms of moles

where n is the number of moles, k = 1.381 x 10-23 J/K is the Boltzmann constant and R = 8.314 J/(mol·K) is the molar gas constant in SI units.

Often it's more convenient not to use the SI units of pressure and volume (Pascals and m3), and to use atmospheres (atm) and liters (L) instead. The gas constant R = 0.08314 L·atm/(mol·K) makes that easy.

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