Gases

Shaun Williams, PhD

The Empirical Gas Laws

Boyle's Law

• Robert Boyle (1627-1691) showed that for a fixed sample of gas at a constant temperature, pressure and volume are inversely proportional
\[ P \propto \frac{1}{V} \] \[ PV = \text{constant} \] \[ P_1V_1 = P_2V_2 \]

Charles' Law

• Charles' Law states that the volume of a fixed sample of gas at constant pressure is proportional to the temperature.
\[ V \propto T \] \[ \frac{V}{T} = \text{constant} \] \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]

Gay-Lussac's Law

• Gay-Lussac's law states that the pressure of a fixed sample of gas is proportonal to the temperature
\[ P \propto T \] \[ \frac{P}{T}=\text{constant} \] \[ \frac{P_1}{T_1}=\frac{P_2}{T_2} \]

Combined Gas Law

• Boyle's, Charles', and Gay-Lussac's Laws can be combined into a single empirical formula
\[ PV \propto T \] \[ \frac{PV}{T}=\text{constant} \] \[ \frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2} \]

Avogadro's Law

• Amedeo Avogadro (1776-1856) did extensive work with gases
• Avogadro found an important relationship regarding the number of moles of gas when the temperature and pressure are constant
\[ n \propto V \] \[ \frac{n}{V}=\text{constant} \] \[ \frac{n_1}{V_1}=\frac{n_2}{V_2} \]

The Ideal Gas Law

• The ideal gas law combines the empirical gas laws into a single expression in terms of the universal gas constant, \(R\)
\[ PV=nRT \]

Note: \( R=N_Ak_B \) where \( k_B = 1.38064852\times 10^{-23}\, \bfrac{J}{K} \) and \( N_A= 6.022140857\times 10^{23}\,mol^{-1} \)

The Kinetic Molecular Theory of Gases

The kinetic molecular theory of gases has 5 postulates in its modern form

1. Gas particles obey Newton’s laws of motion and travel in straight lines unless they collide with other particles or the walls of the container.
2. Gas particles are very small compared to the averages of the distances between them.
3. Molecular collisions are perfectly elastic so that kinetic energy is conserved.
4. Gas particles so not interact with other particles except through collisions. There are no attractive or repulsive forces between particles.
5. The average kinetic energy of the particles in a sample of gas is proportional to the temperature.

Maxwell Distribution of Speeds

• Based on the kinetic molecular theory of gases we can develop an expression of the fraction of gas particles with a specific speed \[ f(v) = 4\pi \sqrt{\left( \frac{m}{2\pi k_BT} \right)^3} v^2 \exp \left( -\frac{mv^2}{2k_BT} \right) \]

Calculating the Average Speed of a Gas

We can calculate the average molecular speed of a gas

Solving the Integral

\[ \left< v \right> = 4\pi \sqrt{\left( \frac{m}{2\pi k_BT} \right)^3} \int_{-\infty}^{\infty} v^3 \exp \left( -\frac{mv^2}{2k_BT} \right)\, dv \]

We can find how to solve this integral in a table of integrals: \[ \int_0^\infty x^{2n+1} e^{-ax^2} \, dx = \frac{n!}{2a^{n+1}} \]

So \[ \begin{eqnarray} \left< v \right> &=& 4\pi \sqrt{\left( \frac{m}{2\pi k_BT} \right)^3} \left[ \frac{1}{2\left( \frac{m}{2k_BT} \right)^2} \right] \\ &=& \left(\frac{8k_BT}{\pi m}\right)^\bfrac{1}{2} \end{eqnarray} \]

Example 2.1

What is the average value of the squared speed according to the Maxwell distribution law?

Note: \( \int_0^\infty x^{2n}e^{-ax^2}\,dx=\frac{1\cdot 3\cdot 5\cdots (2n-1)}{2^{n+1}a^n}\sqrt{\frac{\pi}{a}} \)

Note 2: The root-mean-squared (RMS) speed of gas particle is the squareroot of the average value of \(v^2\) \[ v_{rms}=\sqrt{\left< v^2 \right>} \]

Kinetic Energy

Using expresson for \(v_{mp}\), \(v_{avg}\), and \(v_{rms}\) we can develop expression for the kinetic energy using \[ E_k=\frac{1}{2}mv^2 \]

Real Gases

The van der Waals Equation

• Johannes van der Waals (1837-1923) suggested an modified version of the ideal gas law to account for the deviations of real gases from ideality \[ \left( P+\frac{a}{V_m^2} \right)\left( V_m-b \right) = RT \]

Other Real Gas Laws