Molecular Kinetic Theory

To understand the kinetic theory of gases equation, we first need to consider the molecules of the gas. Each of the molecules will have its own kinetic energy but not all will have the same energy and therefore, may not have the same velocity. This brings the idea of the root mean squared velocity (c_{\text{rms}}). 

The root mean squared velocity (c_{\text{rms}}) is an accurate way of calculating the average speed of all molecules and is the square root of the average of all particles velocity squared. 

The kinetic theory of gases equation states:

PV=\dfrac{1}{3}Nm(c_{\text{rms}})^2

• P= pressure in pascals \text{(Pa)}
• V= volume in cubic metres \text{(m}^3\text{)}
• N= the number of molecules in the gas
• m= the mass of one molecule of  the gas in kilograms \text{(kg)}
• c_{\text{rms}}= the root mean squared speed in metres per second \text{(ms}^{-1}\text{)}

The average kinetic energy of each molecule of a gas in a container can also be calculated using the equation below:

E_k=\dfrac{1}{2}m(c_{\text{rms}})^2=\dfrac{3}{2}kT=\dfrac{3}{2}\dfrac{RT}{N_A}

• E_k= the average kinetic energy in joules \text{(J)}
• m= the mass of each gas molecule in kilograms \text{(kg)}
• k= the Boltzmann constant (=1.38 \times 10^{-23} \text{ kgm}^2\text{s}^{-2}\text{K}^{-1})
• T= the temperature in kelvin \text{(K)}
• R= the molar gas constant (=8.31 \text{ kgm}^2\text{s}^{-2}\text{K}^{-1}\text{mol}^{-1})
• N_A= Avogadro’s number (=6.02 \times 10^{23})

Example: Calculate the average kinetic energy of a molecule of hydrogen-2. The c_{\text{rms}} of hydrogen-2 is 700 \text{ ms}^{-1}.

[3 marks]

 

Firstly we need the mass of a single hydrogen-2 molecule:

1 \text{ mol}= 2 \text{ g} = 0.002 \text{ kg}

Therefore the mass of one molecule:

\dfrac{0.002}{6.02 \times 10^{23}} = \bold{3.32 \times 10^{-27}} \textbf{ kg}

Then substitute values into the equation for energy:

\begin{aligned} \bold{E_k} &= \bold{\dfrac{1}{2}mc_{\text{rms}}} \\ &= \dfrac{1}{2} \times 3.32 \times 10^{-27} \times \textcolor{f21cc2}{700}^2 \\ &= \bold{8.13 \times 10^{-22} \textbf{ J}} \end{aligned}

If very small particles (such as smoke or a powder) are placed into a container of gas or liquid, the particles can be observed via a microscope as moving randomly and erratically. This random motion is known as Brownian motion. 

As the particles move randomly, this observation shows that particles have a range of kinetic energies and no preferred direction of motion. 

As the particles of the gas or liquid are too small to be observed directly, the larger particles need to be added. These larger particles are able to be moved by the smaller particles as they have such high kinetic energies. 

As the larger particles can be seen to collide with the walls of the container, this also acts as evidence for the concept of pressure.