Once winter hits and the air starts getting colder, you might notice your car or bike's tires looking a little deflated. Did your tires spring a leak? While it may seem like it, no air has escaped, it's the pressure that's decreased, not the amount of air itself.
So what happened? Well, this is because of the decrease in temperature. In this article, we will learn about Gay-Lussac's law, which explains the relationship between temperature and pressure for gases.
- This article covers Gay-Lussac's Law
- First, we will do an overview of the components of Gay-Lussac's law (ideal gas, temperature, and pressure)
- Then, we will learn about Gay-Lussac's law and understand what it means
- Next, we will look at the equation that is paired with the law and use it in an example
Gay-Lussac's Law Overview
Before we learn about Gay-Lussac's law, let's do a brief overview of the components involved: ideal gases, temperature, and pressure. First up, let's talk about ideal gases.
When looking at this law and other related gas laws, we are typically applying them to ideal gases, instead of "real" ones:
An ideal gas is a theoretical gas that follows these rules:
- They are constantly moving.
- They have a negligible mass.
- They do not attract or repel other particles.
- They have full elastic collisions (no kinetic energy is lost).
Lastly, let's talk about pressure. Since (ideal) gases are constantly in motion, they often collide with each other and the walls of their container. Pressure is the force of the gas particles colliding with a wall, divided by the area of that wall.
Now that we've learned about the key players, let's talk about Gay-Lussac's Law
Gay-Lussac's Law Statement
Gay-Lussac's Law states that the pressure of a gas (with a given mass and constant volume) will be proportional to the temperature of the gas.
Gay Lussac's Law Explanation
As we talked about earlier, temperature tells us how much kinetic energy a gas has, which in turn tells us how fast it is moving. When temperature increases, the more the gas particles will move around in their container, which increases the likelihood of collisions. Since the number of collisions is increasing, therefore so is the pressure (since more collisions=more force applied).
Here is a diagram to illustrate this:
Fig.1-Diagram of Gay-Lussac's law
On the right-hand side, the gas particles have an initial temperature and pressure (P1). When the temperature increases (shown by an increase in arrows), the pressure also increases (P2). We can see this change by looking at how the number of collisions increases (both wall and particle-particle collisions).
To make sure the law is followed, the mass of the particles and the volume of the container have stayed the same.
Essentially, Gay-Lussac's law states that since the temperature is increasing, the pressure should also increase by a proportionate amount.
In our introduction, we talked about tire pressure decreasing with the weather. When the temperature decreases, the pressure will also decrease. The same quantity of gas is in the tire, but the pressure has decreased, so the tire will appear less "firm" and look deflated.
Gay-Lussac's law equation
Gay-Lussac's law can be expressed mathematically in two different ways.
The first is like this:
$$ P \propto T$$
Where P is pressure, T is temperature, and ∝ is the symbol for "proportional to".
This first equation is essentially just the definition written in mathematical form.
Our next equation requires some slight derivation. First, let's look at a general graph for the law:
Fig.2-Gay-Lussac's law graph
The graph is linear (\(y=mx\)), with the slope (m) being the proportionality constant.
The standard linear equation is y=mx+b, where b is the y-intercept. When the temperature (x) is 0, then there is no movement of gases, meaning that the pressure (y) is also 0. This means that the y-intercept is also 0.
The equation for the graph looks like this:
$$P=mT$$
Or, to put it another way:
$$\frac{P}{T}=m$$
This means that for any given pressure and volume (for an ideal gas), they will have a ratio of m.
Using this equation, we can derive the other form of Gay-Lussac's law:
$$\frac{P_1}{T_1}=m$$
$$\frac{P_2}{T_2}=m$$
$$\frac{P_1}{T_1}=\frac{P_2}{T_2}$$
We can use this equation to solve for the new pressure or temperature of a gas when the other variable has been changed.
The ideal gas law
Gay-Lussac's law can be combined with other gas laws (such as Charles's law and Boyle's law) to create the ideal gas law.
The ideal gas law describes how ideal gases behave, the formula is:
$$PV=nRT$$
Where P is pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is temperature.
We can see where Gay-Lussac's law fits in since P and T are directly proportional (they are on opposite sides of the equation, which indicates direct proportionality).
Gay-Lussac's law examples
Now that we are familiar with the equation, let's put it to use in an example problem.
A 20 L container of nitrogen gas has an initial temperature and pressure of 300 K and 0.8 atm. If the temperature is increased to 425 K, what is the new pressure of the container, assuming the mass and volume stay fixed?
Given the problem, here are the values of our variables:
$$P_1=0.8\,atm\,\,T_1=300\,K\,\,P_2=?\,\,T_2=425\,K$$
So all we need to do is plug these values into our equation and solve for P2:
$$\frac{P_1}{T_1}=\frac{P_2}{T_2}$$
$$T_2*\frac{P_1}{T_1}=P_2$$
$$P_2=(425\,K)*\frac{0.8\,atm}{300\,K}$$
$$P_2=1.13\,atm$$
As an aside, we can also solve for m to calculate either a pressure change or a temperature change (if given the other variable).
In the case above, m would be:
$$\frac{P_1}{T_1}=m$$
$$\frac{0.8\,atm}{425\,K}=0.00188\frac{atm}{K}$$
So what does this mean exactly? Well, if I am given a new temperature, I would multiply it by this factor to get the new pressure. If I was given a new pressure, I would divide it by the factor to get the new temperature.
For example, at 500 K:
$$500\,K*0.00188\frac{atm}{K}=0.94\,atm$$
and at, 1.23 atm:
$$\frac{1.23\,atm}{0.00188\frac{atm}{K}}=654\,K$$
The main point here is that there are several ways you could solve this problem, as long as you follow Gay-Lussac's Law.
Gay-Lussac's Law - Key takeaways
- An ideal gas is a theoretical gas that follows these rules:
- The gas particles are constantly moving.
- The gas particles have a negligible mass.
- The gas particles have negligible volume.
- They do not attract or repel other particles.
- They have full elastic collisions (no kinetic energy is lost)
- Gay-Lussac's Law states that the pressure of a gas (with a given mass and constant volume) will be proportional to the temperature of the gas.
- The formulas for Gay-Lussac's law are:
- $$ P \propto T$$ (Where P is pressure, T is temperature, and ∝ is the proportion symbol)
- $$\frac{P_1}{T_1}=\frac{P_2}{T_2}$$
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