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What is an Ideal Gas?

Important Formula

Formula: Ideal Gas Law

$$\mathit{\Pi} \, V ~=~ n \, R \, T$$ $$\mathit{\Pi} ~=~ \frac{n \, R \, T}{V}$$ $$V ~=~ \frac{n \, R \, T}{\mathit{\Pi}}$$ $$T ~=~ \frac{V \, \mathit{\Pi}}{n \, R}$$ $$n ~=~ \frac{V \, \mathit{\Pi}}{R \, T}$$ $$R ~=~ \frac{V \, \mathit{\Pi}}{ n \, T }$$

Rearrange formula

What do the formula symbols mean?

Pressure

$$ \mathit{\Pi} $$ Unit $$ \mathrm{Pa} = \frac{ \mathrm{N} }{ \mathrm{m}^2 } $$

This pressure is present in a confined system containing an ideal gas. According to the ideal gas law, the pressure increases when the temperature $T$ of the gas increases or the volume $V$ in which the gas is confined decreases.

Volume

$$ V $$ Unit $$ \mathrm{m}^3 $$

The volume of a closed system containing an ideal gas.

Temperature

$$ T $$ Unit $$ \mathrm{K} $$

It is the absolute temperature (in Kelvin) of the gas in a closed system.

Amount of substance

$$ n $$ Unit $$ \mathrm{mol} $$

The amount of substance indirectly indicates the number of gas particles. It is related to the particle number $N$ by the Avogardo constant $N_{\text A}$: $ n = \frac{N}{N_{\text A}} $.

Gas constant

$$ R $$ Unit $$ \frac{\mathrm J}{\mathrm{mol} \, \mathrm{K}} $$

Molar gas constant (also called universal gas constant) is a physical constant from thermodynamics and has the following exact value: $$ R ~=~ 8.314 \, 462 \, 618 \, 153 \, 24 \, \frac{\mathrm J}{\mathrm{mol} \, \mathrm{K}} $$

Table of contents

Important Formula
Exercises with Solutions

Ideal gas is characterized by the following properties:

It consists of point-like particles of the same mass $m$.
There are no molecular forces between the particles.
Particle collisions are elastic.
The velocity direction of the particles is disordered.
The particle's velocity is given by the Maxwell distribution.
The gas is described by the ideal gas equation:

$$ \begin{align} \mathit{\Pi} \, V ~=~ n \, R \, T \end{align} $$

What is a degree of freedom?

The degrees of freedom of a molecule, for example, is the number of possible displacements in space and therefore also the possibility of absorbing energy. A particle that can perform rotational motion in addition to translational motion can also absorb rotational energy.

A monatomic molecule, for example, can move in 3 directions - it therefore has 3 degrees of freedom. Diatomic molecules can also rotate around two axes, which is why they have 5 degrees of freedom. And polyatomic molecules have 6 degrees of freedom - 3 of which are translation and 3 rotation.

Exercises with Solutions

Use this formula eBook if you have problems with physics problems.

Exercise #1: Tire Pressure after a Journey

Before starting your journey, you check the tire pressure of your car. A pressure gauge shows a pressure of $3 \, \text{bar}$ at a temperature of 20°C. After the journey, you find that the tire has warmed up to 60°C.

Assume that the tire has not expanded, and the air inside the tire can be considered as an ideal gas. What is the pressure inside the tire?

Solution to Exercise #1

Use the ideal gas to set up equations for the initial and final states of the tire. Since the tire does not expand according to the exercise, the volume $ V $ remains the same before and after the journey.

Before the journey, the pressure $ \mathit{\Pi}_{\text{before}} = 3 \, \text{bar} $ and the temperature $ T_{\text{before}} = 20 \,^{\circ} \text{C} $. So, the ideal gas equation for this state is: 1 \[ \mathit{\Pi}_{\text{before}} \, V ~=~ n \, R \, T_{\text{before}} \]

After the journey, the pressure has changed to an unknown value $ \mathit{\Pi}_{\text{after}} $, and the temperature has increased to $ T_{\text{after}} = 60 \,^{\circ} \text{C} $. Therefore, the ideal gas equation for the tire state after the journey is: 2 \[ \mathit{\Pi}_{\text{after}} \, V ~=~ n \, R \, T_{\text{after}} \]

The volume $ V $, as well as the amount of substance $ n $ and the gas constant $ R $, are all constants. Therefore, bring them to the right side of the equation and the variable pressure and temperature to the left side. Then, equations 1 and 2 transform to: 3 \[ \frac{\mathit{\Pi}_{\text{before}}}{T_{\text{before}}} ~=~ \frac{n \, R}{V} \] 4 \[ \frac{\mathit{\Pi}_{\text{after}}}{T_{\text{after}}} ~=~ \frac{n \, R}{V} \]

Now, the same constants $ \frac{n \, R}{V} $ are on the right side in both equations. This means you can equate 3 and 4 to get rid of the constants: 5 \[ \frac{\mathit{\Pi}_{\text{after}}}{T_{\text{after}}} ~=~ \frac{\mathit{\Pi}_{\text{before}}}{T_{\text{before}}} \]

Perfect! Now you have three known quantities and only one unknown quantity in the equation, namely the desired tire pressure $ \mathit{\Pi}_{\text{after}} $ after the journey. Simply rearrange 5 with respect to $ \mathit{\Pi}_{\text{after}} $: 6 \[ \mathit{\Pi}_{\text{after}} ~=~ \frac{\mathit{\Pi}_{\text{before}}}{T_{\text{before}}} \, T_{\text{after}} \]

Now just substitute the given values. But BE CAREFUL! Don't forget to first convert the temperature to Kelvin scale, otherwise the result will be incorrect. Also, the pressure is given in bar ($1 \, \text{bar} = 10^5 \, \frac{\text N}{\text{m}^2}$) and not in Pascal ($ 1\, \text{Pa}= 1 \, \frac{\text N}{\text{m}^2} $). With $ 3\, \text{bar} = 300 000 \, \text{Pa} $, as well as $ 20\, ^{\circ}\text{C} = 293.15 \, \text{K}$ and $ 60\, ^{\circ}\text{C} = 333.15 \, \text{K}$, you get the desired tire pressure: 7 \[ \mathit{\Pi}_{\text{after}} ~=~ \frac{ 300 000 \, \text{Pa} }{ 293.15 \, \text{K} } \, 333.15 \, \text{K} ~=~ 340 000 \, \text{Pa} \]

This corresponds to a pressure: $ \mathit{\Pi}_{\text{after}} ~=~ 3.4 \, \text{bar} $.