Deviation From Ideal Gas Law

Ideal Gas Law

Before diving into the definitions and ideal and real gases, let's review about the basic properties of gases and the kinetic molecular theory.

Gases are one of the three most common states of matter. Gases are commonly known for:

• Assuming the shape and volume of a container.

• Being compressible.

• Having random, fast motion of molecules.

If a gas molecule behaves according to the "Kinetic Molecular Theory", then we consider them an ideal gas. When you are dealing with an ideal gas, the ideal gas law can be used to relate its volume, pressure, the number of moles, and temperature.

Pressure, Volume, and Temperature

Pressure is an intensive property, meaning that it does not depend on the amount of matter or substance present. The pressure of a gas is the force exerted by the gas on the surface of its container.

The SI unit of pressure is the pascal (1 Pa = 1 N/m² or 1 kg m-s²) but it can be converted between different units. Some conversions that you might come across are:

• 1 atm = 760 torr.

• 1 atm = 760 mmHg = 1.013x105 Pa = 101.325 kPa.

• 1 atm = 101,325 Pa.

• 1 bar = 0.9869 atm.

• 1 bar = 10⁵ Pa.

• 1 mmHg = 133.322 Pa.

• 1 psi = 6.895 kPa.

Let's look at an example where you need to convert between different units of pressure!

Gases also have partial pressure, which is the pressure exerted by an individual gas within a mixture. Partial pressure is a part of Dalton's Law of Partial pressures, which states that the sum of the partial pressures of each individual gas in a container is equal to the total pressure of the mixture of gases in the same container.

The equation Dalton's Law of Partial pressure is:

$$P_{total}=P_{A}+P_{B}+...$$

Fig. 1: Visualizing Partial Pressure, Isadora Santos - StudySmarter Originals.

If you are starting to feel confused, let's look at an example of how to calculate the total pressure of a container using Dalton's Law!

Volume is an extensive property that also affects the behavior of gases. Extensive properties are properties that depend on the amount of matter being measured. In gases, the volume is not fixed because gases take the shape of the container. Gases can also be compressed when pressure is applied, resulting in a change in volume.

The SI base unit for volume is the liter (L). But, similar to pressure, it can be converted to other units!

• 1 L = 10^-3 m³.
• 1 L = 1 dm³.
• 1 L = 10³ cm³.
• 1 L = 1.0567 qt.
• 1 gallon = 4 qt.
• 1 gal = 3.7854 L.
• 1 cm³ = 1 mL.
• 1 in³ = 16.39 cm³.

So, if you understand how to use these different conversions, you can apply them to various examples like the one below!

Temperature is an intensive property, so it does not depend on the amount of substance. Changing temperature causes phase changes.

You might be asked to convert between other temperature units in your exam, so these are the conversions you should know:

• Temperature from Celsius to Kelvin:

$$T_{K}=T_{Celsius}+273.15$$

• Temperature from Kelvin to Celsius:

$$T_{Celsius}=T_{K}-273.15$$

• Temperature from Fahrenheit to Celsius:

$$T_{Celsius}=\frac{T_{Fahrenheit}-32}{1.8}$$

• Temperature from Celsius to Fahrenheit:

$$T_{Fahrenheit}=(1.8\cdot T_{Celsius})+32$$

There are three laws that, when combined, create the Ideal Gas law Formula. These three laws are Boyle's Law, Charles's Law, and Avogadro's Law:

• Boyle's Law states that a confined gas's volume (V) is inversely proportional to the pressure (P) exerted on the gas.
• Charles' Law states that the temperature of a gas is directly proportional to the volume of a gas.
• Avogadro's Law states that the quantity of gas is directly proportional to the volume of a gas, as long as the pressure and temperature of the gas stay the same.

Fig. 2: Graphical representation of Boyle's law, Charles's Law, and Avogadro's Law, Isadora Santos - StudySmarter Originals.

The formula for the Ideal Gas Law is:

$$P\cdot V=n\cdot R\cdot T$$

Where:

• P = pressure in Pa.
• V = volume of gas in liters.
• n = amount of gas in moles.
• R = universal gas constant = 0.082057 L·atm / (mol·K).
• T = temperature of the gas in Kelvin (K).

Now, let's work through an example and apply the ideal gas law equation!

Density and Deviation from Ideal Gas Law

We can also use the ideal gas law to find the density of gases. Density is also an intensive property and can be calculated by dividing mass over volume. When dealing with density, we say that gases deviate from the ideal gas law at moderate and higher densities.

The ideal gas law formula relating density is:

$$d=\frac{m}{V}=\frac{PM}{RT}$$

Where:

• d = density (g/L).
• P = pressure (atm).
• M = molar mass (g).
• R = ideal gas constant (R = 0.08206 J/mol·K).
• T = temperature (K).

Let's utilize the ideal gas/density formula and solve an example!

Deviations from the Ideal Gas Law

So far, we have only looked at ideal gases and how they behave. Now, we have to consider what happens when gases do not behave ideally! If a gas molecule deviates from ideal behavior (according to the kinetic molecular theory), then we call them real gases. Compared to ideal gases, real gases do have volume and forces of attraction/repulsion present, so particles do interact with one another.

Deviations from the ideal gas law are less at low pressure and high temperature, and greater at high pressure and low temperature. Also, heavier gases tend to deviate most from the ideal gas behavior.

This deviation at high pressure and low temperature happens because, at low temperatures, the attractive forces can interact with the gas molecules because they have low speeds. When gases are at a low temperature, then they will have low kinetic energies.

• Temperature and kinetic energy are directly proportional to one another.

Since real gases deviate from the ideal gas law, what can we do if we needed to calculate the pressure, volume, amount of gas, or temperature of real gases? Easy. We can use van der Waal's equation for real gases!

$$[P+a(\frac{n}{V})^{2}]\cdot [V-bn]=nRT$$

• P = pressure of the gas
• V = volume of gas
• n = amount of gas in moles
• R = universal gas constant = 0.082057 L·atm / (mol·K)
• T = temperature of the gas in Kelvin (K)
• Constant a and Constant b = depend on the type of gas we are dealing with.

The van der Waal's equation was actually derived from the ideal gas law formula, and some corrections were made to consider the volume occupied by the gas and the intermolecular forces that are present. This is what the constants a and b are: correction factors! The values for these constants are specific for each gas. If you come across a question asking to use the real gas formula, you will be given these values, so you worry about memorizing them!

So, try not to feel overwhelmed when you see this equation! It's not as complicated as it looks and can be easily applied in calculations.

Boyle's Law Deviation from Ideal Gas

Let's talk about Boyle's law! This law states that, at a constant temperature and number of moles, the volume (V) of a confined gas is inversely proportional to the pressure (P) exerted on the gas In other words, when volume increases, pressure decreases (and vice versa).

When we say that real gases deviate from the ideal gas law, it means that real gases do not follow Boyle's law, Charles's law, or Avogadro's law perfectly!

Conditions and Deviations from the Ideal Gas Law

To make things simple, let's summarize the conditions for deviation from the ideal gas law!:

• Gases only obey the ideal gas law at high temperature and low pressure. So, gases will deviate from the ideal gas law at high pressure and low temperatures.
• Ideal gases are assumed to occupy a negligible volume. But, in reality, gas molecules do occupy volume, especially at low temperatures and high pressures!
• Ideal gases are said to have perfect elastic collisions. Now we know that at low temperature and high pressure, gas pressures are lower than predicted and do possess some intermolecular interactions.

Now, you should have an understanding of ideal and real gases, and also the different equations that are used to analyze their behavior!