Combined Gas Law — Overview & Calculations

Combined Gas Law

What is the Combined Gas Law?

• The Combined Gas Law states that for a fixed amount of gas: $\frac{PV}{T}=$ k
  (k being the proportionality constant)

• The Combined Gas Law combines Charles's Law, Boyle's Law, and Gay-Lussac's Law

• The Combined Gas Law is written as: $\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$

Combined Gas Law Formula Cheat Sheet!

How to effectively use the combined gas law

1) Write down all of the variables given by the problem

2) Identify what the unknown is

3) Solve for the unknown by plugging into the combined gas law

(or quickly solve by plugging into the correct formula from the cheat sheet!)

Combined Gas Law Practice Problem Walkthrough

A gas has an initial pressure of 400 atm, a temperature of 40 K, and a volume of 200 mL. The volume is then increased to 300 mL and the pressure is decreased to 200 atm, what is the new temperature?

1) Write down all of the variables given

$P_{1}$ = 400 atm

$P_{2}$ = 200 atm

$V_{1}$ = 200 mL

$V_{2}$ = 300 mL

$T_{1}$ = 40 K

$T_{2}$ = ?

2) Identify the unknown

In this case, we are looking for $T_{2}$, so we can either plug in our variables to the combined gas law equation and solve for $T_{2}$, or we can use the cheat sheet to directly solve for $T_{2}$

3) Solve for the unknown

$T_{2}$ = $\frac{P_2T_1V_2}{P_1V_1}$

$T_{2}$ = $\frac{\left(200\right)\left(40\right)\left(300\right)}{\left(400\right)\left(200\right)}$

$T_{2}$ = 30 K

Now try a practice problem!

Practice problem solution

(Video) Combined Gas Law

by Tyler DeWitt

This video will teach you how you can combine all of the gas laws you've learned into one law: $$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$ This one formula is very useful and you can change it in whatever way you need to solve a problem. You will also get a chance to practice using this law with an example problem.

We've been studying the behavior of gases. The kinetic molecular theory explained the assumption we make about gas molecules. For example, there is a lot of empty space between the molecules. So, they don't experience any of Van der Waal's forces. It also helped us imagine the gas's movement in a container. They move in a straight line until they collide with another molecule or the sides of the container. Once we understood their behavior, we turned our attention to the gas laws.

We started with Boyle's law. It inversely linked the gas pressure and volume. Next, we turned to Charles' law. It examined the direct relationship between a gas's volume and temperature. Finally, we studied Gay-Lussac's law. It compared the pressure and temperature of a gas.

The gas laws helped us explore the relationships between volumes, pressures, and temperatures. But, what happens if we need to consider all three? We use the combined gas law! The combined gas law blends Boyle's, Charles', and Gay-Lussac's laws. So, now we can solve problems where volume, pressure, and temperature change! Let's explore the combined gas law.

Forms of the Combined Gas Law

We have two rules for the combined gas law. The first is that we assume the moles of gas stay constant. The second is that all temperature measurements are in Kelvin. Just like the other gas laws, the combined gas law has multiple forms. Let's explore each one.

• $\frac{PV}{T}=k$

The first form is a ratio that equals a constant. We multiply pressure and volume in the numerator. In the denominator, we have the temperature. The combined gas law is difficult to use qualitatively. For example, if the temperature increases, the product of pressure and volume increases. But, that does not tell us much. For example, we don't know if the pressure increased by one percent or ninety-nine percent. The same is true of the volume's increase.

• $\frac{P_{1}V_{1}}{T_{1}}=\frac{P_{2}V_{2}}{T_{2}}$

The power of the combined gas law comes from the second form. In form one, the ratio was a constant. So, if we change all the variables, the proportion is still the same. Therefore, the initial is equal to the final. If a system's pressure, volume, and temperature change, we can still calculate each term.

Using the Combined Gas Law

There are two ways to use the combined gas law. The easiest way is to derive the Boyle's, Charles' or the Gay-Lussac law. If the temperature is kept constant, the combined gas law reverts to $P_{1}V_{1}=P_{2}V_{2}$. That's Boyle's law! When we fix the pressure, we have $\frac{V_{1}}{T_{1}}=\frac{V_{2}}{T_{2}}$ which is Charles' law. Finally, if we keep the volume the same, we get the Gay-Lussac law. The equation becomes $\frac{P_{1}}{T_{1}}=\frac{P_{2}}{T_{2}}$.

The second way to use the combined gas law is to solve for one of the variables. For example, let's imagine we had one mole of the noble gas helium. It's at standard temperature and pressure. Now, we heat the gas to 300.0 K. The pressure increases to 1.5 bar. What's the new volume?

First, list all the terms.

At STP, $P_{1}=1.0 bar$ and $T_{1}=273.15 K$.

The problem does not tell us the initial volume. We need to remember that one mole of gas at STP is 22.4 liters. So, $V_{1}=22.4$.

The problem tells us that the ending pressure and temperature are $P_{2}=1.5 bar$ and $T_{2}=300.0 K$.

Let's solve for the final volume.

$\frac{P_{1}V_{1}}{T_{1}}=\frac{P_{2}V_{2}}{T_{2}}$

$\frac{1.0 bar\times{22.4 L}}{273.15 K}=\frac{1.5 bar\times V_{2}}{300.0K}$

$0.082=0.005\times V_{2}$

$V_{2}=16.4 L$

Combined Gas Law at Absolute Zero

Just like Charles' law, the combined gas law breaks down as we approach absolute zero. We know that any gas would condense before reaching absolute zero. As we decrease the temperature, we lessen kinetic energy. When the kinetic energy is low, the molecules experience electrostatic interactions. So, they behave as real gases. The combined gas law is a good approximation under standard conditions.

Adding Moles to the Combined Gas Law

The combined gas law assumes that the moles are constant. What happens if we change the number of moles too? We have the ideal gas law! It adds Avogadro's relationship to the other gas laws. From the ideal gas law, you can derive all the connections, including the combined law.