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# Ideal gases

## Ideal gases

An ideal gas is a theoretical gas composed of non-interacting point particles. This implies that individual particles have zero volume and that intermolecular forces amongst them are negligible.

Real gases (like oxygen in the air) have non-zero volume and intermolecular forces. Nonetheless, at low pressures they often behave like ideal gases.

Under low pressure, the distance between gas particles is very large relative to the volume of the gas. In addition, the intermolecular forces become negligible due to the separation of particles.

The ideal gas concept is a therefore a good approximation for gases under at low pressures.

This is important because it is much easier to calculate the volume, pressure and temperature of an ideal gas (as compared to a real gas).

The particles of an ideal gas have zero volume and do not interact with each other. They only exert pressure on the container.

## Ideal gas equations

The ideal gas equation gives the relationship between the pressure, volume and temperature of an ideal gas. An alternative definition of an ideal gas is one that respects this equation given by:$$pV=nRT=NkT$$$p=$pressure of the gas; $V=$volume of the gas; $n=$number of particles of the gas measured in moles; $N=$number of particles, $R=$molar gas constant; $k=$Boltzmann constant; $T=$thermodynamic temperature of the gas.

The volume of individual particles does not matter in the calculation of the volume of the gas (this arises from the definition of ideal gas particles as point particles).

Note that the quantities and constants must be expressed in SI units.

## Constants and quantities used in the ideal gas equations

One mole is equivalent to $6.02\times10^{23}$ particles. This value is known as the Avogadro constant ($N_{\text{A}}$). It was obtained by determining the number of atoms in $0.012\text{ kg}$ of carbon-12.

The number of particles $N$ is related to the number of moles $n$ by:$$N=nN_{\text{A}}$$

The molar gas constant $R$ is equivalent to $8.314\text{ m}^{2}\text{ kg}\text{ s}^{-2}\text{ K}^{-1}$. It is a commonly used universal constant in thermodynamic calculations.

The Boltzmann constant $k$ is equivalent to $1.38\times10^{-23}\text{m}^{2}\text{kgs}^{-2}K^{-1}$. It is related to the molar gas constant $R$ by:$$R=kN_{\text{A}}$$

## Mean kinetic energy of an ideal gas particle

The mean kinetic energy of a gas particle $(\langle e_{k} \rangle )$ of mass $m$ is given by:$$\langle e_{k} \rangle=\frac{1}{2}m\langle v^{2}\rangle=\frac{3}{2}kT$$ $\langle v^{2}\rangle$ is the mean square speed of the particle.

The mean kinetic energy is directly proportional to the thermodynamic temperature $T$.

It is necessary to take the mean square speed (as opposed to the mean speed squared $(\langle v\rangle)^{2}$) because the mean speed ($\langle v\rangle$) of a particle is zero.

This is because the particles within a gas move in random directions and the probability that a particle moves in a particular direction is equal to that of any other direction.

This means that on average, each particle is considered to be stationary (i.e. $\langle v\rangle=0$). This can be inferred from the fact that a container filled with gas does not move even though the gas particles are in perpetual motion.

The mean kinetic energy equation is derived by equating the macroscopic (i.e. using the ideal gas equation) and microscopic (i.e. considering the force of a particle on a wall) pressure values of a gas particle.

$p=$pressure of the gas; $V=$volume of the gas; $n=$number of particles of the gas measured in moles; $k=$Boltzmann constant; $T=$thermodynamic temperature of the gas.

## Mean kinetic energy of one mole of an ideal gas

The mean kinetic energy of one mole of gas is equivalent to the mean kinetic energy of one gas particle multiplied by Avogadro's constant $N_{\text{A}}$. This is given by:$$\frac{1}{2}mN_{\text{A}}\langle v^{2}\rangle=\frac{3}{2}kN_{\text{A}}T\Rightarrow \frac{1}{2}m_{\text{r}}\langle v^{2}\rangle=\frac{3}{2}RT$$$m_{\text{r}}$ (or $M$) is the molar mass (i.e. mass of one mole) of the gas. It is the product of Avogadro's constant and the mass of one particle (i.e. $m_{r}=N_{A}m$).

The internal energy of an ideal gas is equivalent to its mean kinetic energy (recall that an ideal gas has negligible intermolecular forces and thus zero potential energy):

$$U=\frac{3}{2}NkT$$

The internal energy is directly proportional to the thermodynamic temperature $T$.

$k=$Boltzmann constant; $m=$mass of a particle; $n=$number of particles of the gas measured in moles; $N=$number of particles, $R=$molar gas constant; $T=$thermodynamic temperature of the gas; $U=$internal energy of the gas.

## Root mean square speed of an ideal gas

The root mean square speed ($v_{\text{rms}}$) of the particles of a gas is the root of the mean of the squares of the individual speeds of the particles. It is a type of "average" speed of the gas particles.

The mean velocity of gas particles within a stationary container would be zero as there is no effective movement of the gas as a whole.

The root mean square speed gives a meaningful summary of the speed of the particles of a gas and is easily used in calculations.

The root mean square speed is given by:$$v_{\text{rms}}=\sqrt{\langle v^{2}\rangle}=\sqrt\frac{3kT}{m}=\sqrt\frac{3RT}{mN_{\text{A}}}=\sqrt\frac{3RT}{m_{\text{r}}}$$

$k=$Boltzmann constant; $m=$mass of a particle; $m_{r}=$molar mass; $R=$molar gas constant; $T=$thermodynamic temperature of the gas.