Gas Laws

Gases are very interesting in that the quantities of pressure, volume, and temperature change how gases fundamentally behave in given systems. For a solid or liquid, its volume is usually rigidly defined and so is its mass so it's hard to observe extreme changes to these phases of matter when you change these quantities.

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While gases are easily mutable, it's also important to understand just exactly quantities like volume, pressure, temperature, and mass affect gases. These effects have been modeled over the course of history in many experiments and below is a scientific reference to them.

Boyle's Law states that for a given mass of gas at a constant temperature, its pressure and volume are inversely proportional.

This proportionality can be intuitively understood with a very simplistic example. If we take gas at constant mass and temperature and double its volume, its pressure will be cut in half and vice versa.

In terms of the properties of ideal gases, this has to do with the fact that the pressure a gas exerts with its containment system has to do with how frequently it collides with the container walls. If it collides more frequently, it will exert more force on the container walls, which means it will exert greater pressure. If you increase the volume and keep temperature and mass constant, the distance between container walls will increase so it will take longer for gas molecules to go between container walls. Since the time increases, the frequency of collisions will decrease and so will the pressure.

This law was discovered by Robert Boyle in 1662 but French physicist Edme Mariotte also independently discovered it in 1679, so the law can also be called Mariotte's Law, or the Boyle-Mariotte Law.

Above is the mathematical expression for Boyle's Law.

Charles's Law states that for a given mass of an ideal gas at constant pressure, the volume of the gas is directly proportional to the absolute temperature in kelvins(K).

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This law was determined in 1787 by Jacques Charles for closed systems.

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In terms of an example, if we double the volume of gas, this means we also double its absolute temperature in Kelvins(K).

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In terms of the properties of ideal gases, this law can be conceptually explained by the fact that increasing the volume requires the average kinetic energy of the molecules, which is directly proportional to their temperature, to be greater to keep the pressure equal. You can think of it like this. The pressure of a gas is determined by how frequently it collides with the container walls. If the volume of the container increases, the gas molecules must move faster between collisions for the frequency to stay the same. If the gas molecules move faster, their temperature will then increase in these conditions.

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This law was determined in 1787 by Jacques Charles for closed systems.

Above is the mathematical expression for Charles's Law.

Gay-Lussac's Law states that for a given mass of an ideal as at constant volume, the pressure of the gas exerted on the walls of its container is directly proportional to the gas's absolute temperature in kelvins(K).

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In terms of an example, if we double the pressure of a gas, this means we also double its absolute temperature(in kelvins).

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In terms of the properties of an ideal gas, this law can be conceptualized by the fact that that if a given mass of gas in a container is rigid, then an increase in its pressure, or frequency of collisions, will be accompanied by an increase in its temperature, or average kinetic energy. Think of it like this. If I increase the pressure exerted by the gas, then the gas molecules must be hitting the walls at a much quicker rate. However, the distance between walls is constant so that means for the collisions to be more frequent, the molecules must be faster. If they were faster, then they'd zoom about faster and collide more. Since this increased average kinetic energy is directly proportional to the temperature of the molecules, that means this concept can be expressed as a direct relationship between pressure and absolute pressure.

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This law was officially determined in 1808 by Joseph Louis Gay-Lussac but he figured it out sometime between 1800 and 1802 whilst building an air-based thermometer. However, the law would accurately be more accredited to Guillaume Amontons, who discovered it almost a century prior. Therefore, Gay-Lussac's Law usually refers to the law of combining volumes but it can still refer to this given proportionality.

Above is the mathematical expression for Gay-Lussac's Law.

Avogadro's Law states that the volume of an ideal gas is directly proportional to the number of gas molecules in the container.

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An example of this proportionality is that if you tripled the gas molecules in a container and kept the pressure and temperature constant, the container's volume would also triple.

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In terms of the properties of the gas, we can think of it like this. Imagine if we had a container filled with gas and had a pump of gas connected into it. Since we are analyzing the change in volume, imagine the container isn't rigid, like a balloon. If we pump gas molecules into the balloon and the pressure and temperature are constant, the balloon will expand. Exhibit A for this phenomenon would be when people blow party balloons. They push air into the balloon and the balloon's temperature stays mainly constant and for low volumes, the pressure doesn't play too big of a role.

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This law was named after Amadeo Avogadro, who hypothesized this law in 1812. It's no surprise that a gas law with moles is named after Avogadro, the pioneer behind the mole itself.

Above is the mathematical expression for Avogadro's Law.

One thing to notice is that all of the gas laws above are mathematically expressed similarly. They deal with two quantities in an initial state(denoted by subscript 1) having the same proportion in a final state(denoted by subscript 2), making these quantities state variables. A state variable is a variable that can determine the states of a system and is unaffected by what happens between those states. Pressure and volume only depend on what happens at a final state and what happens at an initial state. An example of a state function is displacement. You could travel all the way across the entire world and come back to the same spot, which would make your net displacement 0. However, you could also achieve this just by standing in place. The fact that the displacement is independent of what path you took and thus, what happened in between, means that it only depends on a given state and is a state variable.

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Now, all of these proportionalities are great but what if we had multiple changes to a system and it wasn't as simple as a two-variable proportion? Well, the combined gas law capitalizes on this by combining the proportionalities in Boyle's, Charles's, and Gay-Lussac's Laws, giving the relationship between the pressure, volume, and absolute temperature of a gas at a constant mass.

If we combine Avogadro's Law, we get the finalized Ideal Gas Law:

​where R is a constant of proportionality known as the Universal Gas Constant and P is the pressure, V is the volume, n is the number of moles of gas in the given volume, and T is the absolute temperature.

If we multiply the number of moles by Avogadro's number, we get the number of molecules. We can apply that to this equation and re-express it in terms of molecules of gas. However, since we multiplied by Avogadro's Number, we also divide the gas constant by that to give us this modified equation.

where k_b is Boltzmann's constant, which is just the universal gas constant divided by Avogadro's Number.

A modified form of the Ideal Gas Law is using M_m, the molar mass, and lower-case rho, the density, is given below.

STP(Standard Temperature and Pressure) are the standard conditions for experiments to be made at.

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This should not be confused with standard state for thermodynamics.

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At STP, the temperature is 0 C(273 K)and the pressure is 1 atm.

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Because we know the pressure and temperature of a gas at STP are constant, if we solve for the volume per mole, we get this:

At STP, the volume per unit mole of a gas is always 22.4 L/mol. This "constant" allows for many gas-based and stoichiometric calculations to become simplified.