Wednesday, November 25, 2009

IDEAL GASES

The Ideal Gas Law

The Ideal Gas Law

If you only glean one scrap of knowledge from this SparkNote, make sure that it is the ideal gas law equation:

PV = nRT

This is the meat and potatoes of gases. With it, you will be able to solve almost any gas equation involving the pressure, volume, amount, and temperature of a gas.

Before we jump in, though, we need to get some basics down. The first two sections of this topic lay the foundation for the ideal gas law. Section oneintroduces Boyle's law and the manometer. Both measure the volume and pressure of a gas. Section twointroduces Charles' law and Avogadro's law. Charles' law relates the temperature and volume of a gas. Avogadro's law relates the quantity a gas and its volume.

Boyles', Charles', and Avogadro's laws combine to form the ideal gas law, which is the uber law of gases. In the third section you'll see why. The ideal gas law can be manipulated to explain Dalton's law, partial pressure, gas density, and the mole fraction. It can also be used to derive the other gas laws. In short, it will satisfy most of your gas-based needs.

Let us address one caveat before we begin. The ideal gas law is an ideallaw. It operates under a number of assumptions. The two most important assumptions are that the molecules of an ideal gas do not occupy space and do not attract each other. These assumptions work well at the relatively low pressures and high temperatures that we experience in our day to day lives, but there are circumstances in the real world for which the ideal gas law holds little value. With this in mind, let us begin.


Terms and Formulae

Terms
Absolute temperature - A temperature scale whose lowest possible value is zero. Absolute temperature is measured in Kelvin.
Absolute zero - A temperature where T = 0K . The theoretical lowest possible temperature.
Avogadro's law - Avogadro's law relates the amount and volume of a gas at constant temperature and pressure. Mathematically:

fracVn = k

k is a constant unique to the temperature and pressure.
Avogadro's number - N A = 6.022×1023 . An avogadro's number of molecules equals one mole.
Boyle's law - A gas law relating pressure and volume for a fixed amount of gas at a constant temperature. Mathematically:

PV = C

C is a constant unique to the amount of gas and temperature.
Charles' law - A gas law relating volume and temperature for a fixed amount of gas at constant pressure. Mathematically:

= k

k is a constant unique to the amount of gas and pressure. Note that T must be an absolute temperature(in Kelvins).
Dalton's law - The total pressure of a mixture of gases is the sum of the pressures each constituent gas would exert alone. Mathematically:

P tot = P A + P B + P C + ƒƒƒ

Gas constant - Constant R in the ideal gas law. The value of R varies with the units of P , V , n , and T . The value of R can be deduced from the following table:
UnitsValue of R


0.08206


8.314


8.314


1.987


62.36
Ideal gas law - A gas law stating that PV = nRT . The two main assumptions of the law are that the molecules of an ideal gas do not have volume and do not interact with each other. The ideal gas law is a good approximation when the pressure is low and the temperature is high.
Isothermal conditions - Two or more conditions that share the same temperature. In other words, T is constant.
Kelvin - A unit of absolute temperature. Abbreviated with the letter "K." The Kelvin scale is related to the Celsius scale by T K = T C + 273.15 . Kelvin should be used for all classical and ideal gas law calculations.
Manometer - A device used to measure the difference in pressure between two gases:
"A" and "B" represent the atmosphere, a vacuum, or a pressurized gas.
Molar mass - The mass of one mole of particles. Commonly expressed as g/mol.
Mole - One mole contains Avogadro's number ( 6.022×1023 ) of particles. For example, one mole of H 2 would contain 6.022×1023 H 2 molecules. Moles are abbreviated as "mol."
Mole fraction - In a mixture of gases, the ratio that relates the number of moles of a constituent gas to the total number of moles in the mixture. Derived using the mole fraction formula.
Partial pressure - In a mixture of gases, the pressure exerted by one constituent gas. The sum of the partial pressures of gases in a mixture is equal to the total pressure of the mixture.
Standard atmospheric temperature and pressure - Conditions where T = 298K and P = 1bar .
Standard temperature and pressure (STP) - Conditions where T = 273K andP = 1atm .
Formulae
Boyle's law formula

PV = C

C is a constant unique to the amount of gas and temperature.
Charles' law formula

= k

k is a constant unique to the amount of gas and pressure. Note that T must be an absolute temperature.
Dalton's law formula

P tot = P A + P B + P C + ƒƒƒ

Gas density formula

d ==

Ideal gas law formulaPV = nRT
Kelvin âÜî Celsius conversionT K = T C + 273.15

Boyle's Law and the Manometer

Boyle's Law

The most important thing to remember about Boyle's Law is that it only holds when the temperature and amount of gas are constant. A state of constant temperature is often referred to as isothermal conditions. When these two conditions are met, Boyle's law states that the volume V of a gas varies inversely with its pressure P . The equation below expresses Boyle's law mathematically:

Mole fraction formula
PV = C

C is a constant unique to the temperature and mass of gas involved. plots pressure versus volume for a gas that obeys Boyles law.
Figure %: Pressure versus Volume

You will get the most mileage out of another incarnation of Boyle's law:

P 1 V 1 = P 2 V 2

The subscripts 1 and 2 refer to two different sets of conditions. It is easiest to think of the above equation as a "before and after" equation. Initially the gas has volume and pressure V 1 and P 1 . After some event, the gas has volume and pressure V 2and P 2 . Often you will be given three of these variables and asked to find the fourth. You should realize that this is a simple case of algebra. Separate the knowns and unknowns on two different sides of the "=" sign, plug in the known values, and solve for the unknown.

The Manometer

Boyle used a manometer to discover his gas law. His manometer had an odd "J" shape:

Figure %: A Manometer
As you can see from , there are two ends to Boyle's manometer. One end is open to the atmosphere. The other end is sealed, but contains gas at atmospheric pressure. Since the pressure on both ends of the tube is the same, the level of mercury is also the same.

Next Boyle added mercury to the open end of his manometer.

The volume of the gas at the closed end of the manometer decreased, but since gas can't get in or out of the closed end, the amount of gas does not change. Likewise we can assume that the experiment occurs under isothermal conditions. Boyle's law should hold, meaning that the initial volume times pressure should equal the volume times pressure after the additional mercury was added. Let's use the equation below on the gas at the sealed end:

P 1 V 1 = P 2 V 2

The pressure of the gas before mercury is added is equal to the atmospheric pressure, 760 mm Hg (let's assume that the experiment is run at o C so that 1 torr = 1 mm Hg). So P 1 = 760 mm Hg. The volume V 1 is measured to be 100 mL.

After Boyle added mercury, the volume of the gas, V 2 , drops to 50 mL. To find the value of P 2 , rearrange the equation above and plug in values:


P 2=P 1 V 1/V 2
=(100 mL)(760 mm Hg)/(50 mL)
=1520 mm Hg

If you look back at , you'll notice that the difference P 2 - P 1 = 760 mm Hg, and that this exactly equals the difference in mercury levels on the two sides, h . In fact, Boyle's manometer illustrates a truism common to all manometers: h corresponds to the difference in pressure between the two ends of the manometer.

Boyle's manometer is only one of the many kinds of manometers you'll face. Don't be disheartened; all manometers are practically the same. Realize that each end of a manometer can only be:

  • sealed and contain a vacuum ( P = 0 )
  • open to the atmosphere ( P = P atm )
  • open to a sample of gas with pressure P
This is the key to solving manometer problems. Once you figure out the pressure at both ends of the manometer, you can use the difference to determine the height h of the liquid column, and vice versa.

Let's try this procedure with a manometer in which one end is open to the atmosphere (760 mm Hg) and the other is sealed off to a vacuum.

At the end that is sealed off with a vacuum, P = 0 mm Hg. At the end open to the atmosphere, P = 760 mm Hg. The difference between the two pressures is 760 mm Hg, so the height h must correspond to 760 mm Hg, the atmospheric pressure. Thus this manometer has the same function as a barometer; it measures atmospheric pressure.

There are a few other flavors of manometer, but you can handle them if you remember that h is the pressure difference between the two sides of the manometer. Note that the side of the manometer with the highest pressure also has the lowest level of Hg.


Charles, Avogadro, and the Ideal Gas Law

Charles' Law

Charles' law states that, at a constant pressure, the volume of a mixed amount of gas is directly proportional to its absolute temperature:

= k

Where k is a constant unique to the amount of gas and pressure. Just as with Boyle's law, Charles' law can be expressed in its more useful form:

=

The subscripts 1 and 2 refer to two different sets of conditions, just as with Boyle's law.

Why must the temperature beabsolute? If temperature is measured on a Celsius (non absolute) scale,T can be negative. If we plug negative values of T into the equation, we get back negative volumes, which cannot exist. In order to ensure that only values of V≥ 0 occur, we have to use an absolute temperature scale where T≥ 0 . The standard absolute scale is the Kelvin (K) scale. The temperature in Kelvin can be calculated via T k = T C + 273.15 . A plot of the temperature in Kelvin vs. volume gives :

Figure %: Temperature vs. Volume
As you can see from , Charles' law predicts that volume will be zero at 0 K. 0 K is the absolutely lowest temperature possible, and is called absolute zero.

Avogadro's Law

Avogadro's law states that the volume of a gas at constant temperature and pressure is directly proportional to the number of moles of gas present. It's mathematical representation follows:

fracVn = k

k is a constant unique to the conditions of P and T . n is the number of moles of gas present.

1 mole (mol) of gas is defined as the amount of gas containing Avogadro's number of molecules. Avogadro's number ( N A ) is

N A = 6.022×1023

1 mol of any gas at 273 K (0_C) and 1 atm has a volume of 22.4 L. The conditions 273 K and 1 atm are the standard temperature and pressure (STP). STP should not be confused with the less common standard atmospheric temperature and pressure (SATP), which corresponds to a temperature of 298 K and a pressure of 1 bar.

The numbers 22.4 L, 6.022×1023 , and the conditions of STP should be near and dear to your heart. Memorize them if you haven't already.

The Ideal Gas Law

Charles', Avogadro's, and Boyle's laws are all special cases of the ideal gas law:

PV = nRT

T must always be in Kelvin. n is almost always in moles. R is the gas constant. The value of R depends on the units of P , V and n . Be sure to ask your instructor which values you should memorize.
UnitsValue of R


0.08206


8.314


8.314


1.987


62.36
You can think of R as a converter that changes the units on the right side of the above equation to the units on the left side of the "=" sign. The values 0.0821and 8.314 get the most use. Memorizing them will make your life easier.

The ideal gas law is the equation you must memorize for gases. It not only allows you to relate P , V , n and T , but can replace any of the three classical gas laws in a pinch. For example, let's say you're given constant values of P and n , but forget how Charles' law relates V and T . Rearrange the ideal gas law to separate the constants and unknowns:

= = k

Voila! We have derived Charles' law from the ideal gas law. n , R , and T are constants, so is just the constant k from Charles' law.

The ideal gas law is also useful for those rare occasion when you forget the value of a constant. Let's say I forgot the value of R in . If I remember that a mole of gas has a volume of 22.4 L at STP (760 torr, 273 K), I can rearrange PV = nRT to solve for R in the desired units. It is much more efficient to memorize the values, but it is comforting to know that you can always fall back on good old PV = nRT .

Applying the Ideal Gas Law

Ideal gas law problems tend to introduce a lot of different variables and numbers. The sheer amount of information can be confusing, and it is wise to develop a systematic method to solve them:

1) Jot down the values of P , V , n , and T . If the question says that one of these variables is constant or asks you to find the value of one or the other, make a note of it. Every time you encounter a numerical value or variable, try to fit it into your PV = nRT scheme.

2) Rearrange PV = nRT such that the unknowns and knowns are on opposite sides of the "=" sign. Make sure that you are comfortable with the algebra involved.

3) Convert to the appropriate units. Generally you'll want to deal with SI units ( m 3 , Pa, K, mol). There will be times that non-SI units will be more convenient. In these cases, remember that T must always be in Kelvin. Make sure to select the correct value and units of R .

4) Plug in values and solve for the unknown(s). Ideal gas problems involve a great deal of algebra. The only way to master this type of problem is to practice. Use the problems provided at the end of this section and your textbook until the manipulations of PV = nRT become familiar.

5) Take a step back and check your work. The easiest way to do this is to carry all of the units through your ideal gas calculations. When you're about to solve the equation, make sure that the units on both sides of the "=" sign are equivalent. For simpler problems, it is also worthwhile to make sure that your answer makes sense. For example, if n , R , and T are constant and P rises, make sure that V decreases. It only takes a few seconds, and can save you from some embarrassing mistakes. The usefulness of such commonsense checks decreases as the questions get more complex. For any problem where more than two variables change, you're better off trusting the ideal gas law and your own algebra.

The best advice I can give you is to practice. The more problems you do, the more comfortable you will be with the ideal gas law.

Further Application of the Ideal Gas Law: Dalton's Law, Densities, Mixtures, and Partial Pressure

Gas Density

PV = nRT is an equation, and it can be manipulated just like all other equations. With this in mind, let's see how the ideal gas law can help us calculate gas density.


Density d has the units of mass over volume. The ideal gas law transforms into a form with units in mol per unit volume:

=

generally has the units of mol per liter. If we multiply both sides of the equation by the molar mass of the gas, μ , we get:

d = =

As we can see from this equation, the density d of a gas depends on P , μ , and T . Think about how density will change when the temperature and pressure rise.

Partial Pressure and Mole Fraction

Dalton's law states that the total pressure of a mixture of gases is the sum of the pressures each constituent gas would exert if it were alone. Dalton's law can be expressed mathematically:

P tot = P A + P B + P C + ...

Each individual pressure P A , P B , P C , etc. is the pressure exerted by each constituent gas A, B, or C. P A is called the partial pressure of gas A.

Each individual gas obeys the ideal gas law, so we can rearrange PV =nRT to find pressure:

P A = n a

Since gases A, B, and C are all in the same mixture, they all have the same temperature and volume. P tot also has the same temperature and volume. When P Ais placed over P tot , the variables T , R , and V cancel to give the following result:

=

The quantity is called the mole fraction of gas A and is abbreviated ρ A .

Dalton's law problems often present two containers of gas, mix them, and ask you to find the partial pressures of each gas. There's usually an easy way and a hard way to do such problems; the trick is finding the easy way. You'll gain this intuition quickest if you jump right in. Try your hand at the problems at the end of this section and in your textbook.


== X

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