Ideal Gas: Recap

Just as important as knowing these equations is knowing that they only apply to ideal gases!

An ``ideal'' gas is one with point-like, non-interacting molecules. However the molecules are allowed to be poly-atomic, and so have internal degrees of freedom (rotational, vibrational). The behaviour of all gases tends to that of an ideal gas at low enough pressures; at STP noble gases such as argon are very close to ideal, and even air is reasonably approximated as ideal.

Ideal gases obey the ideal gas law

$\mbox{\large\colorbox{yellow}{\rule[-3mm]{0mm}{10mm} \ $\displaystyle PV=nRT\qquad\hbox{or}\qquad PV=Nk_{\scriptscriptstyle B}T $ }}$

where $N$ is the number of molecules, $n=N/N_A$ is the number of moles (not to be confused with the number density, $N/V$, also denoted by $n$), $R=8.314 \rm JK^{-1}$ and $k_{\scriptscriptstyle B}=R/N_A=1.381\times10^{-23} \rm JK^{-1}$ is Boltzmann's constant. The ideal gas law encompasses Boyle's Law and Charles' Law. It requires the temperature to be measured on an absolute scale like Kelvin's.

Ideal gases have internal energies which depend only on temperature: if $C_V$ is the heat capacity at constant volume,

$\mbox{\large\colorbox{yellow}{ \parbox{14cm}{ \begin{eqnarray*} E= E(T)\qquad\h... ...ow E\!\!\!&=&\!\!\!C_V T \qquad\hbox{if $C_V$ is constant.} \end{eqnarray*}}}}$

In general the heat capacity may change with temperature; however at STP it is usually adequate to consider it as constant and equal to ${\textstyle \frac 1 2}n_f R$ per mole, where $n_f$ is the number of active degrees of freedom. For monatomic gases $n_f=3$ (translational) and for diatomic gases $n_f=5$ (translational and rotational; vibrational modes are ``frozen out''.)

The heat capacities at constant pressure and at constant volume differ by a constant for ideal gases:

$\mbox{\large\colorbox{yellow}{\rule[-3mm]{0mm}{10mm} \ $\displaystyle C_P-C_V= nR. $ }}$

During reversible adiabatic compression or expansion of an ideal gas the pressure and volume change together in such a way that

$\mbox{\large\colorbox{yellow}{\rule[-3mm]{0mm}{10mm} \ $\displaystyle PV^\gamma = \hbox{constant}\qquad\hbox{where}\qquad \gamma\equiv{C_P\over C_V} $ }}$

For a monatomic gas at STP, $\gamma=5/3=1.67$; for a diatomic gas, $\gamma=7/5=1.4$ . Using the ideal gas law, we also have

$$TV^{\gamma-1} = \hbox{constant}\qquad\hbox{and }\qquad TP^{\frac 1 \gamma-1} = \hbox{constant.}$$

Note that $\gamma-1=nR/C_V$.

There are two ``less ideal'' gases sometimes considered. One is a gas of hard spheres, where we no longer neglect the size of the molecules, but still neglect other interactions. Most of the ideal gas results still hold, but with $V$ replaced by $V-nb$, where $b$ is the ``excluded volume'', the minimum volume taken up by a mole of the molecules. For instance, the equation of state is $P(V-nb)=nRT$.

The other is the van der Waals gas, which also allows for the ``van der Waals'' interaction between neutral molecules which arise when a fluctuating dipole moment in one molecule induces a dipole in another molecule, and the two attract with a force with falls off as $1/r^6$, with $r$ the separation. This attraction reduces the pressure at a given temperature and volume, by an amount which is proportional to $1/V^2$ (or $1/d^6$, where $d$ is the average separation). The equation of state for a van der Waals gas is

$$\left(P+{a n^2\over V^2}\right)(V-nb)=nRT.$$