### 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 be *absolute*? 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 :

### 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} = 6.022×10^{23} |

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, 6.022×10^{23}, 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.

Units | Value of R |
||

| 0.08206 | ||

| 8.314 | ||

| 8.314 | ||

| 1.987 | ||

| 62.36 |

*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 0.0821 and 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.