Real gases don’t always follow the ideal gas law, which is a rule we use to understand how gases behave. Here are some reasons why real gases can act differently, especially when we do math with them:

1. Intermolecular Forces: The ideal gas law assumes that gas molecules don’t affect each other. But in real life, gas molecules can attract or push each other away. This can happen a lot when the pressure is high or the temperature is low.

2. Volume of Particles: The ideal gas law treats gas particles as if they don’t take up any space at all. However, real gas particles do have volume. At high pressures, the space they take up can really matter.

3. Conditions of Use: The differences in behavior become clearer when conditions aren’t perfect. That’s when you should think about using a different equation called the Van der Waals equation:

   $\left(P + \frac{a}{V^2}\right)(V - b) = nRT$

   Here, $a$ and $b$ are special numbers for each gas that help us get better results.

When you do calculations with real gases, always check the conditions first. Real gas behavior can mess with your results if you aren't careful!