# Avogadro’s Hypothesis

Hi, and welcome to this video on **Avogadro’s Principle**.

Let’s start with a familiar scenario. Imagine you are inflating a balloon. As you exhale, **molecules** of nitrogen, oxygen, and carbon dioxide push into the balloon, expanding it to a new volume, let’s say 1 liter.

Now, imagine you inflate a second balloon, but this time you use helium to fill the balloon to 1 liter.

We can easily observe that these two balloons are the same size. But it’s much harder to answer why they have the same volume. Are they filled with the same number of particles? Does it matter that one balloon is filled with helium while the other is a mix of molecules? After all, helium atoms are much smaller and lighter than the molecules in air. Or is that all irrelevant?

This was the question **Amedeo Avogadro** was interested in. He studied the relationship between the volume of a gas and the number of particles. And in 1811, he concluded that “equal volumes of all gases, at the same temperature and pressure, have the same number of molecules.” This is Avogadro’s Principle.

So, applying this to our two balloons, we can now say that because both are 1 liter in size and are at the same temperature and pressure, they must contain the same number of particles. This means that the sum of all the nitrogen, oxygen, and carbon dioxide molecules in the first balloon is equal to the number of helium atoms in the second balloon.

nN_{2}+ nO_{2}+ nCO_{2} = nHe

And Avogadro said that this can be extended to any gas! So if we filled a third and fourth balloon to 1 liter with ozone and butane, there would be the exact same number of particles in all four balloons.

Let’s just take a minute to think about it. It’s pretty amazing- especially when you consider how different these substances are. Take for example butane and helium. A single molecule of butane is much larger than a single helium atom. Butane boils at -1 ºC while helium boils at -269 ºC. Furthermore, butane is a highly combustible compound and helium is almost completely inert. And yet, in their gaseous state, they take up the same amount of space.

So what does this mean about substances in the gas phase?

Using Avogadro’s Law, along with other gas laws, scientists deduced that the macroscopic behavior of gases is not significantly influenced by the chemical identity of the particles. This led to the development of the well-known **Ideal Gas Law**, pressure times volume equals the number of moles times the universal gas constant times temperature. The conclusion was also made that we could use it for real gases, not just the fictional ideal gas.

Let’s consider this further. Remember, an ideal gas is a hypothetical substance in which the particles have no **volume** and do not interact. Under standard conditions, we can approximate real gases to be ideal. In other words, real gas particles are so much smaller than the overall volume that it’s okay to assume they take up no space. And because particles are moving so fast and spend so little time next to each other, their interactions are quite insignificant. Thus, we can apply the Ideal Gas Law to any gaseous substance. While we won’t get into the details here, there are circumstances when real gases deviate significantly from the Ideal Gas Law. That’s all we’ll say for now- but just keep this in mind.

For example, we can use the Ideal Gas Law to determine the volume of 1 mole of any gas at standard conditions (0 ºC and 1 atmosphere). Dividing both sides by pressure, we get that the volume is equal to the number of particles times the gas constant times temperature divided by pressure.

Plugging in our values, we get 1 mole times 0.082 liter atmospheres per mole Kelvin times 273 Kelvin divided by 1 atmosphere equals 22.4 liters. As this represents the “standard” volume of 1 mole of gas, 22.4 liters is a rather famous volume and containers of this size are often found in high school chemistry classrooms.

Now, returning to our original 1 liter balloons. Pause the video and take a minute to use the ideal gas equation to solve for the number of particles in each balloon assuming a pressure of 1 atmosphere and a temperature of 23 degrees celsius.

Got it? Let’s do it together now.

\(n=\frac{PV}{RT}=\frac{1\text{ atm }× 1\text{ L}}{0.082\frac{\text{ L atm}}{\text{mol K}}× 296\text{ K}}\)The number of particles equals pressure times volume divided by the gas constant times temperature. Plugging in our values, we have 1 atmosphere times 1 liter divided by the quantity 0.082 liters atmospheres per mole Kelvin times 296 Kelvin, which gives us 0.041 moles of gas. And hopefully you didn’t do this calculation for each balloon because it would have been exactly the same! Note that nowhere in the equation do we include any information about the chemical qualities of the specific gas. So from this one calculation, we know that there were 0.041 moles of gas in all four balloons.

Circling back to Avogadro, his observation that any gas at a constant temperature and pressure would occupy the same volume, helped scientists understand the nature of gaseous particles and what assumptions we could make about their behavior and interactions. This helped develop the kinetic theory of gases, which describes the macroscopic behavior of gases on a microscopic level, as well as the Ideal Gas Law.

In fact, if we rearrange the gas law to solve for volume, we find Avogadro’s Law (the equation behind his Principle).

\(V=\frac{nRT}{P}\)Volume is equal to the moles times the gas constant times temperature divided by pressure. We know R is a constant and we’ve mentioned that Avogadro’s Principle applies only when temperature and pressure are constant. So, under constant temperature and pressure, volume is equal to the moles of a gas times a constant, k, where the constant is equal to R times T divided by P.

V=n × k, where k = \(\frac{RT}{P}\)

In other words, the volume of a gas is directly proportional to the number of particles present – the primary conclusion of Avogadro’s Principle.

Thanks for watching, and happy studying!