The ideal gas law deals with the macroscopic properties of a gas such as temperature, pressure, and volume. The theory behind how gases behave at a microscopic level is called the Kinetic Molecular Theory and has the following postulates:
1. A gas consists of small molecules moving in a straight line until they hit the edges of the container they're in.
2. The individual gas molecules occupy no volume.
3. When gas molecules hit each other, they bounce off each other.
4. The gas molecules are neither attracted to nor repelled by each other.
5. The energy of a gas depends only on the overall temperature of a gas and nothing else.
An important thing to note about these postulates is that they are not 100% accurate. The purpose of these postulates is to provide a model that we can use to predict how gases will behave 99% of the time. Let's go through each postulate individually:
1. A gas consists of small gas molecules moving in a straight line until they hit the edges of the container they're in.
Gas particles don't just move around randomly. If you've taken physics, you can interpret this as the particles obeying Newtonian motion. The particles move in a straight line until they either collide with the edge of the container or until they collide with another gas molecule at which point they both bounce off each other.
2. The individual gas molecules occupy no volume.
A single gas molecule is so small that we model the individual gas molecules as volume-less objects. Even in your room, there is an almost incomprehensible amount of gas molecules floating around, but even the total volume occupied by all of the molecules is still negligible. This is why we model gas molecules as volume-less objects; in reality, they just occupy an infinitesmally small volume.
3. When gas molecules hit each other, they bounce off each other.
This ties into the 1st postulate, which states that gases travel in a straight line until they collide. When two gas molecules collide with each other, they bounce off each other as if they were two perfect rubber balls. In the language of physics, we'd say that the molecules have collisions that are perfectly elastic.
4. The gas molecules are neither attracted to nor repelled by each other.
Recall that all objects have some level of attraction & repulsion through the existence of Van der Waals interactions. Gases exhibit these interactions as well, though the attractive forces are so miniscule that we ignore them for most cases.
5. The energy of a gas depends only on the overall temperature of a gas and nothing else.
This is probably the most difficult to understand intuitively. The energy of a gas is dependent only on the temperature of the gas. No matter what volume or pressure the gas is at, the energy of the gas will always be the same for a certain temperature. We say that the energy of the gas is temperature-dependent.
The way we calculate the energy is through the following formula:
`KE=3/2*k_B*T`
Where `k_B` is the Boltzmann constant, `1.38*10^(-23) ("m"^2*"kg")/("s"^2*K)`, and `T` is the temperature in `K`
Neither the identity of the gas nor its pressure and volume matter when calculating the energy of the gas. The only property that matters is its temperature.
At any given temperature of a gas, the molecules are not all travelling with exactly the same speed. Some of the molecules will be travelling faster whereas others will be travelling slower. If we plot the number of molecules vs. speed, we get what's called a Boltzmann Distribution:
On this graph, the different colored curves represent different temperatures given by the key on the top right corner. The y-axis, `n`, is the number of molecules at the corresponding temperature. For the red curve at `-100°C`, the number of molecules travelling at `(0 m)/s` is almost 0, whereas the number of molecules travelling at `(300 m)/s` is high.
Notice that as the temperature increases from `-100°C rArr 20°C rArr 600°C`, the curve flattens out. For the blue curve at `600°C`, the distribution of molecules at different speeds is much more even than that of the lower temperature curves. Additionally, the number of molecules at higher velocities is greater for higher temperatures. We can generalize these observations through the following statements:
1. Gas molecules at a certain temperature travel within a range of velocities.
2.The higher the temperature of a gas, the larger the range of velocities, and the faster the average speed fo the molecules.
Dalton's Law of Partial Pressures describes the individual contributions to the total pressure of a gas when the gas is a mixture of different gases. Imagine you had a gas of composed of `1/3 A`, `1/4 B`, and `(7)/12 C` where `A`, `B`, and `C` are different gases. If the total pressure is `1 "atm"`, how much of the total pressure is contributed from `A`, `B`, and `C`?
It turns out that `A` contributes to `1/3`, `B` contributes to `1/4`, and `C` contributes to `7/12` of the total pressure. This is Dalton's Law of Partial Pressure, which is commonly stated as:
In a mixture of gases, the partial pressure of each gas is equal to its mole fraction.
The mole fraction (`X`) of a gas refers to the number of moles of the gas divided by the total number of moles. For example, the mole fraction of `A` is calculated as:
`X_A = n_A/n_("total") = (1/3 "moles A")/((1/3 "moles A")+(1/4 "moles B")+(7/12 "moles C"))= 1/3`
If we want to find the partial pressure of gas `A`, then we just multiply the total pressure by the mole fraction of `A`:
`P_A= P_"total" X_A = (1 "atm")(1/3)=1/3 "atm"`
This means that gas `A` contributes to `1/3` of the total pressure of the entire gas. Dalton's Law written mathematically is therefore:
`P_i=P_"total" X_i`
Where `i` is any gas.
A simpler way to think about Dalton's Law is that, in a mixture of gases, the contributions to the total pressure of the gas is the mole fraction of each individual component. If you have a gas consisting of 50% `A` and 50% `B`, then gas `A` will contribute to 50% of the total pressure and gas `B` will contribute to 50% of the total pressure.
1. Kinetic Molecular Theory models a gas as a collection of volumeless particles bouncing around.
2. The energy of a gas depends solely on the temperature of the gas. The identity of the gas, volume, and pressure do not matter.
3. The higher the temperature of a gas, the faster the average gas molecule is moving.
4. A Boltzmann Distribution is a graphical representation of the distribution of velocities for gas particles at a temperature.
5. Dalton's Law of Partial Pressures states that the partial pressure of each gas in a mixture is equal to its mole fraction multiplied by the total pressure.