To recap, we just talked about the foundational Kinetic Molecular Theory (KMT) of gases, which describes gases as large numbers of tiny particles in constant, random motion and is based on five assumptions, and some of the notable gas laws developed by chemists throughout history.

Let’s recount the gas laws we’re talked about so far:

1. Boyle’s Law: $P_1V_1=P_2V_2$
2. Charles’ Law: $\frac{{ V_1}}{{T_1}} = \frac{{ V_2}}{{T_2}}$
3. Gay-Lussac’s Law: $\frac{{ P_1}}{{T_1}} = \frac{{ P_2}}{{T_2}}$
4. Combined Gas Law: $\frac{{P_1 V_1}}{{T_1}} = \frac{{P_2 V_2}}{{T_2}}$
5. Avogadro’s Law: $\frac{{V_1}}{{n_1}} = \frac{{ V_2}}{{n_2}}$

If you want to refresh more about these specific laws and work on a couple examples, you’re more than welcome to check the previous study guide that goes much into detail!

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🎈 The Ideal Gas Law (PV = nRT)

The ideal gas law is like a VIP pass to predicting how gases behave under different conditions. It's an equation that chemists adore because it relates pressure (P), volume (V), number of moles (n), temperature (T), and includes a special constant known as R (the ideal gas constant).

When do gases exhibit ideality, then? Gases are considered ideal at very high temperatures and very low pressures as both conditions minimize interactions between molecules and let them run free without taking up much space. As you lower temperature and increase pressure, you’ll see more and more deviations from ideal behavior in gases of interest.

In real life, ideal gases are hard to come by, but they are useful models we can use to understand how gases respond to changing conditions (that is, pressure, temperature, number of moles, and volume). Before we model real gases, ideal gases are good starting points when it comes to learning gas behaviors. 💙

What makes the ideal gas law special is that it combines the simple gas laws, especially Boyle’s, Charles’, and Avogadro’s) together. Instead of having to memorize five equations, you just need one equation to relate four variables and a constant together! Isn’t that an amazing package deal? 🎉

Practice Questions

❓ Practice Question 1: A sample of neon has a volume of 12 L at sea level (1 atm) and room temperature (25°C). Calculate the number of moles of neon.

Let’s take a look at what we have:

• Volume = 12 L ✅
• Pressure = 1 atm ✅
• Temperature = 25°C ✅
• Gas constant = we always have that ✅
• Moles = missing! 🤔

We, then readjust our Ideal Gas Law to look for moles by isolating it from the other variables:

To keep units consistent, we also convert °C to K: $25°C + 273 = 298\,K$. Now, we’re ready! Plug values into the equation:

The sample of neon, therefore, has 0.49 mol given 12 L at sea level and room temperature. 👏

❓ Practice Question 2: A gas sample contains 0.5 moles of gas at a pressure of 2.50 atm and a temperature of 300 K. Calculate the volume occupied by the gas.

Our givens:

• n = 0.5 moles
• P = 2.50 atm
• T = 300 K
• R = always given

We’re missing V! Rearranging the Ideal Gas Law equation to find V gives us $V = \frac{nRT}{P}$.

Plugging in our numbers:

The gas, therefore, occupies 4.92 L.

What about missing temperatures? Let’s check the next problem:

❓ Practice Question 3: A gas sample contains 2.5 moles of gas at a volume of 10.0 L and a temperature of 400 K. Calculate the pressure of the gas.

Our givens:

• n = 2.5 mol
• V = 10.0 L
• T = 400 K
• R = always given

Rearranging our equation to find P: $P = \frac{nRT}{V}$

Plugging in our numbers:

Last but not least, missing temperatures:

❓ Practice Question 4: A gas sample contains 0.25 moles of gas at a pressure of 2.00 atm and a volume of 5.00 L. Calculate the temperature of the gas.

Our givens:

• n = 0.25 mol
• P = 2.00 atm
• V = 5.00 L
• R = always given

Rearranging our equation to find T: $T = \frac{PV}{nR}$

Plugging in our numbers:

As you probably noticed in the practice problems, as we look for certain variables, the calculations slightly change to adapt for what we’re searching for, so it’s important to keep track of your variables and correctly rearrange the ideal gas equation to suit your needs! 💯

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🌍 Applying the Ideal Gas Law in Our World

If you’re ever skeptical on how the Ideal Gas Law can be applied to daily life because of how abstract it may seem, feel free to take a look at some of the example industries and applications you might see the Ideal Gas Law in action! 🥊

1. Environmental Science: We can use the Ideal Gas Law to calculate air density changes with altitude—super useful when it comes to understanding weather patterns or pollution dispersion throughout the year!
2. Vehicular Engineering: Did you know that airbags need precise calculations so they inflate properly during car crashes? This is all thanks to PV = nRT ensuring safety standards. Phew!

❓ Practice Question 4: What is the density of dry air at STP if dry air has an average molar mass of 29 g/mol?

STP conditions: $P=1\,atm;T=273\,K; R=0.08206\frac{L•atm}{mol•K}$

In this example, we are given average molar mass, pressure, temperature, and R. We find volume:

From here, we can use volume to find density through the relationship $d=\frac{mass}{volume}$, where dry air’s molar mass is around 29 g/mol:

Well done!

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💡 Real Gases & Deviations from Ideality

This content is optional and is intended to satisfy your curiosity!

Sometimes, gases are like rebellious teens—they don't always follow the rules laid out by the ideal gas law, especially under extreme pressures or chilly temperatures.

• When gases are squeezed together (low volumes) or cooled down (low temperatures), they start feeling each other's presence through attractions or repulsions; these are intermolecular forces that cause real gases to deviate from ideality.
• Under high pressures, gas molecules can't ignore their own volume anymore—it becomes significant compared to the space they're in.

Van der Waals Equation – A Closer Look at Reality

To correct for these pesky deviations, scientists came up with the van der Waals equation—a more accurate prediction tool for real gases that factors in particle size b and intermolecular forces a.

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⭐ Wrapping Up the Ideal Gas Law

When gases operate at very high temperatures and very low pressures, they are considered ideal. In that case, we can use the Ideal Gas Law to relate pressure (P), volume (V), number of moles (n), and temperature (K) and predict one of these parameters should we need to calculate them. The Ideal Gas Law is a useful model to look into gas behavior at various environmental conditions before moving on to more complicated models and equations such as the Van der Waals equation.

But for Honors Chemistry, all you need to know is: when in doubt, PV = nRT! ⛽