**The ideal gas Law**

Where does this come from?

**Robert Boyle found**

That is, the product of the pressure of a gas times the volume of a gas is a constant for a given sample of gas. In Boyle's experiments the Temperature (T) did not change, nor did the number of moles (n) of gas present. So Boyle found

but did not explore the effect the temperature, or the number of moles would have on pressure and volume.

**Jaques Charles found**

That is, the volume of a given sample of gas increases linearly with the temperature if the pressure (P) and the amount of the gas (n) is kept constant. So Charles found

**Avagadro's Postulate**

At the same temperature and pressure equal volumes of all gasses contain the same number of molecules.

**V** = **n** (RT/P)

Guy Lussac found that 1 volume of Cl_{2} combined with 1
volume of H_{2} to make 2 volumes of HCl. The equation for
the reaction is

With this example we can clearly see the relationship between the number of moles of a gas, and the volume of a gas.

At constant temperature andpressurethevolumeof a gas is directlyproportionalto the number ofmolesof gas.

Not so coincidentally if V is constant instead of P then

At constant temperature andvolumethepressureof a gas is directlyproportionalto the number ofmolesof gas.

You could remember all the different gas laws,

P

V

and so on...

Or you could think about the problem a bit and use PV=nRT.

N_{2}O is placed in a piston. Initially the volume of the
piston is 3.0 L, and the pressure of the gas is 5.0 atm. The piston
is used to compress the gas to a volume of 1.5 L; determine the
pressure of the N_{2}O.

well, before the compression

or

after expansion

since n, R, and T do not change

substituting

P = 1.0 x 10 L

See, if you forget all those different relationships you can just use PV=nRT.

**A PV = nRT problem**

What is the volume of 1 mole of an ideal gas at STP (Standard Temperature and Pressure = 0 °C, 1 atm)?

(1) V = 1(0.08206)(273.15)

V = 22.41 L

So, the volume of an ideal gas is 22.41 L/mol at STP.

This, 22.4 L, is probably the most remembered and least useful number in chemistry.

What is the volume of 5.0 g NH3 at 25 °C and 1 atm. of pressure?

Well we just found that the volume of 1 mole of an ideal gas is 22.41 L so we can use this as a conversion factor...right?

Everyone remembers that 1 mol of an ideal gas occupies a volume of 22.4 L, but this is probably the least useful number in chemistry. Alot of people forget that this relationship isonlytrue atSTP(0 °C and 1 atm.).

So, use

To use PV=nRT we need to have moles of NH_{3}.

It is not practical to use PV=nRT as a conversion in a factor label problem so we will just solve for V.

V = 7.18 = 7.2 L
NH_{3}

Seltzer water is made by dissolving CO_{2} in water.
Seltzer can be made at home using small containers of pressurized
CO_{2}. If one of the cartridges contains 20.00 mL
CO_{2} at 55.00 atm at 23.0 °C and it expands into an
empty seltzer bottle with a volume of 1.000 L and the resulting
pressure is 1.000 atm what is the temperature of the gas.

Before the gas expands...

and

after the gas expands...

or

since R has not changed (it is called the universal gas
CONSTANT for a reason) and we have not
changed the number of moles of CO_{2}...

or...

continuing the previous problem

What will the pressure be when the gas warms to 23.0 °C?

before expansion and warming...after expansion and warming to room temp

once again R and n have not changed, and now not even T has not changed so...

solve for P