Solving IF problems algebraically

Let's solve this problem algebraically:
A child is holding a 5.00 L helium-filled balloon. The temperature of the balloon is 294 K and the pressure inside the balloon is 0.966 atm. If the child accidentally lets go of the balloon and it floats up to a place high in the sky where the temperature is only 261 K and the pressure drops to 0.78 atm, what will be the new volume of the balloon?

We can organize the information from the problem in a table:

	initial	final
P	0.966 atm	0.78 atm
V	5.00 L	?
T	294 K	261 K

We start with the basic algebraic statement:

It is worth noting that temperatures are rarely recorded in Kelvin, but we'll deal with that later.

In this case, amount is never mentioned, so we'll eliminate that from the formula.

Then we'll solve it for V_f

Substituting in our values allows us to solve the problem:

Here's another example with much more annoying units.
A 5.40 L balloon holds 10.0 g of carbon dioxide at 34^oC and 1.06 atm. If 0.227 moles of carbon dioxide are added to the balloon, the pressure changes to 1500 mm Hg and the temperature drops to -17.2^oC, what will the new volume be?

	initial	final
P	1.06 atm	1500 mm Hg
V	5.40 L	?
T	34oC	-17.2oC
n	10.0 g CO2	+ 0.227 mols

Before we can solve this problem, we need to deal with these units.
First, temperature MUST be in Kelvin, so we'll need to change both temperatures by adding 273.15 to each value.

	initial	final
P	1.06 atm	1500 mm Hg
V	5.40 L	?
T	307 K	256.0 K
n	10.0 g CO2	+ 0.227 mols

The pressures must be in the same unit. Although either unit will work, we'll convert the mm Hg to atmospheres.

We also need to figure out the values for amount. First we'll convert the mass of CO₂ to moles

then we'll add the 0.227 moles. This will leave us with this data table:

	initial	final
P	1.06 atm	1.97 atm
V	5.40 L	?
T	307 K	256.0 K
n	0.227 mol	0.454 mol

Now, we're ready to solve the problem. Starting with the general formula:

There is nothing to eliminate here (all variables are being used), so we can rearrange this to solve for V_f: