Avogadro's gas law states the volume of a gas is proportional to the number of moles of gas present when the temperature and pressure are held constant. This example problem demonstrates how to use Avogadro's law to determine the volume of a gas when more gas is added to the system.

### Avogadro's Law Equation

Before you can solve any problem regarding Avogadro's gas law, it's important to review the equation for this law. There are a few ways to write this gas law, which is a mathematical relation. It may be stated:

k = V/n

Here, k is a proportionality constant, V is the volume of a gas, and n is the number of moles of a gas. Avogadro's law also means the ideal gas constant is the same value for all gases, so:

constant = p_{1}V_{1}/T_{1}n_{1}= P_{2}V_{2}/T_{2}n_{2}

V_{1}/n_{1}= V_{2}/n_{2}

V_{1}n_{2}= V_{2}n_{1}

where p is pressure of a gas, V is volume, T is temperature, and n is number of moles.

### Avogadro's Law Problem

A 6.0 L sample at 25°C and 2.00 atm of pressure contains 0.5 mole of a gas. If an additional 0.25 mole of gas at the same pressure and temperature are added, what is the final total volume of the gas?

### Solution

First, express Avogadro's law by its formula:

V_{i}/n_{i}= V_{f}/n_{f}

where

V_{i}= initial volume

n_{i}= initial number of moles

V_{f}= final volume

n_{f}= final number of moles

For this example, V_{i} = 6.0 L and n_{i} = 0.5 mole. When 0.25 mole is added:

n_{f}= n_{i}+ 0.25 mole

n_{f}= 0.5 mole = 0.25 mole

n_{f}= 0.75 mole

The only variable remaining is the final volume.

V_{i}/n_{i}= V_{f}/n_{f}

Solve for V_{f}

V_{f}= V_{i}n_{f}/n_{i}

V_{f}= (6.0 L x 0.75 mole)/0.5 mole

V_{f}= 4.5 L/0.5 V_{f}= 9 L

Check to see if the answer makes sense. You would expect the volume to increase if more gas is added. Is the final volume greater than the initial volume? Yes. Doing this check is useful because it is easy to put the initial number of moles in the numerator and the final number of moles in the denominator. If this had happened, the final volume answer would have been smaller than the initial volume.

Thus, the final volume of the gas is 9.0

### Notes Regarding Avogadro's Law

- Unlike Avogadro's number, Avogadro's law was actually proposed by Amedeo Avogadro. In 1811, he hypothesized two samples of an ideal gas with the same volume and at the same pressure and temperature contained the same number of molecules.
- Avogadro's law is also called Avogadro's principle or Avogadro's hypothesis.
- Like the other ideal gas laws, Avogadro's law only approximates the behavior of real gases. Under conditions of high temperature or pressure, the law is inaccurate. The relation works best for gases held at low pressure and ordinary temperatures. Also, smaller gas particles—helium, hydrogen, and nitrogen—yield better results than larger molecules, which are more likely to interact with each other.
- Another mathematical relation used to express Avogadro's law is:

V/n = k

Here, V is the volume, n is the number of moles of the gas, and k is the proportionality constant. It's important to note this means the ideal gas constant is *the same* for all gases.