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## Calculations for relationships between volume and number of moles of a gas.

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How much air do you put into a tire?

A flat tire is not very useful. It does not cushion the rim of the wheel and creates a very uncomfortable ride. When air is added to the tire, the pressure increases as more molecules of gas are forced into the rigid tire. How much air should be put into a tire depends on the pressure rating for that tire. Too little pressure and the tire will not hold its shape. Too much pressure and the tire could burst.

You have learned about Avogadro’s hypothesis: equal volumes of any gas at the same temperature and pressure contain the same number of molecules. It follows that the volume of a gas is directly proportional to the number of moles of gas present in the sample. Avogadro’s law states that the volume of a gas is directly proportional to the number of moles of gas when the temperature and pressure are held constant. The mathematical expression of Avogadro’s law is

$V=k \times n \quad \text{and} \quad \frac{V_1}{n_1}=\frac{V_2}{n_2}$

where  $n$ is the number of moles of gas and  $k$ is a constant. Avogadro’s law is in evidence whenever you blow up a balloon. The volume of the balloon increases as you add moles of gas to the balloon by blowing it up.

If the container holding the gas is rigid rather than flexible, pressure can be substituted for volume in Avogadro’s law. Adding gas to a rigid container makes the pressure increase.

A balloon has been filled to a volume of 1.90 L with 0.0920 mol of helium gas. If 0.0210 mol of additional helium is added to the balloon while the temperature and pressure are held constant, what is the new volume of the balloon?

Step 1: List the known quantities and plan the problem.

Known

• $V_1=1.90 \text{ L}$
• $n_1=0.0920 \text{ mol}$
• $n_2=0.0920+0.0210=0.1130 \text{ mol}$

Unknown

• $V_2=? \text{ L}$

Note that the final number of moles has to be calculated by adding the original number of moles to the moles of added helium. Use Avogadro’s law to solve for the final volume.

Step 2: Solve.

First, rearrange the equation algebraically to solve for $V_2$ .

$V_2=\frac{V_1 \times n_2}{n_1}$

Now substitute the known quantities into the equation and solve.

$V_2=\frac{1.90 \text{ L} \times 0.1130 \text{ mol}}{0.0920 \text{ mol}}=2.33 \text{ L}$

Since a relatively small amount of additional helium was added to the balloon, its volume increases slightly.

#### Summary

• Calculations are shown for relationships between volume and number of moles of a gas.

#### Practice

Work the problems at the site below:

#### Review

1. What is held constant in the Avogadro’s Law relationship?
2. What happens if you add gas to a rigid container?
3. Why does a balloon expand when you add air to it?