High School Chemistry/Using Algebra in Chemistry

During your studies of chemistry (and physics also), you will note that mathematical equations are used in a number of different applications. Many of these equations have a number of different variables with which you will need to work. You should also note that these equations will often require you to use measurements with their units. Algebra skills become very important here!

Lesson Objectives

 * Be able to rearrange mathematical formulas for a specific variable.
 * Have an understanding of how to use units in formulas.
 * Be able to express answers in significant figures and with units.

Solving Formulas with Algebra
Sometimes, you will have to rearrange an equation to get it in the form that you need. When you are working with an equation such as $$D = \tfrac{M}{V}$$ (density = mass/volume) and asked to solve for density, it is relatively easy; all you have to do is substitute the measurements and solve – of course, keeping in mind significant figures!

If you are asked to solve the above equation for M, then you will need to manipulate the equation to isolate the desired variable, in this case in the form of "M = ". To do this, you will need to move the V from the right side to the left side of the equation. As the V is in the denominator, you will need to multiply both sides of the equation by V:


 * $$V \times D = (V)\left(\frac{M}{V}\right)$$

Multiplying this out:


 * $$V \times D = M\,\!$$

A similar process is used if you need to solve for V:

Take the previous equation (V × D = M) and divide both sides by D:


 * $$\left ( \frac{V \times D}{D} \right ) = \left ( \frac{M}{D} \right )$$

Solving, this becomes:


 * $$V = \frac{M}{D}$$

What if you are given a more complex equation, like PV = nRT, and asked to solve for n? You need to follow the same steps as you did in the above examples. The only difference is that there are more symbols to rearrange.

Look at the original equation: PV = nRT. Our goal is to get n on one side of the equation by itself.

To remove the RT from the right side, we will divide both sides by RT:


 * $$\frac{PV}{RT} = \frac{nRT}{RT}$$

After solving, we are left with:


 * $$n = \frac{PV}{RT}$$

Algebra with Units and Significant Figures
So far, you've learned about units, significant figures, and algebraic manipulation of equations. Now it's time to put all three of these together. We’ll start with a simple example: density. Density is a measure of the amount of mass per unit of volume, and a common unit used is g/mL. In the first example, we're going to do a straightforward calculation of density from a given mass and volume.

What is the density of an object that has a mass of 13.5 g and a volume of 7.2 mL?

Solution:

The equation is as follows:


 * $$D = \frac{\text{mass}}{\text{volume}}$$

Substituting in the known values (with units):


 * $$D = \frac{13.5\,\text{g}}{7.2\,\text{mL}}$$

Finally, solving the equation and rounding off the answer based in significant figures:


 * $$D = \frac{13.5\,\text{g}}{7.2\,\text{mL}} = 1.9\,\text{g/mL (2 significant figures)}$$

This calculation was easy to do because there was no rearranging of the equation and no cancellation of units.

For the next example, let's look at a more complex calculation.

A sample of an ideal gas has a volume of 14.2 L at a pressure of 1.2 atm. If the gas pressure is increased to 1.8 atm, what is the new volume?

Solution:

This problem uses Boyle's law:


 * $$P_1 V_1 = P_2 V_2\,\!$$

All the variables are known except for V2, so the equation needs to be rearranged to solve for the one unknown. We can do this by dividing both sides by P2:


 * $$\frac{P_1 V_1}{P_2} = V_2$$

Now we substitute in the known values and their units:


 * $$\frac{(1.2\,\text{atm})(14.2\,\text{L})}{(1.8\,\text{atm})} = V_2$$

Next, we cancel out units:


 * $$\frac{(1.2)(14.2\,\text{L})}{(1.8)} = V_2$$

Finally, we calculate our answer and round off to the appropriate number of significant figures:


 * $$V_2 = 9.5\,\text{L}\,\!$$

Lesson Summary

 * Students of chemistry need to be able to use algebra in their calculations.

Review Questions

 * 1) For the equation PV = nRT, re-write it so that it is in the form of "T = ".
 * 2) The equation for density is $$D = \tfrac{M}{V}$$. If D is 12.8 g/cm3, and M is 46.1 g, solve for V, keeping significant figures in mind.
 * 3) The equation P1 V1 = P2 V2, known as Boyle's law, shows that gas pressure is inversely proportional to its volume. Re-write Boyle's law so it is in the form of V1 = ?.
 * 4) The density of a certain solid is measured and found to be 12.68 g/mL. Convert this measurement into kg/L.
 * 5) In a nuclear chemistry experiment, an alpha particle is found to have a velocity of 14,285 m/s. Convert this measurement into miles/hour.