Ideal and Real Gases – Quick Review

Gaseous State Basics

• Only 11 common elements exist as gases at ambient conditions: $\text{H}<em>2, \text{He}, \text{N}</em>2, \text{O}<em>2, \text{F}</em>2, \text{Ne}, \text{Cl}_2, \text{Ar}, \text{Kr}, \text{Xe}, \text{Rn}$
• Key traits: highly compressible, exert uniform pressure, low density, no fixed shape/volume, mix completely without agitation.

Fundamental Gas Laws

• Boyle: $v \propto \frac{1}{p}\;(T,n \text{ constant}) \;\Rightarrow\; pv = \text{const}$
• Charles: $v \propto T\;(p,n \text{ constant}) \;\Rightarrow\; \frac{v}{T}=\text{const}$
• Avogadro: Equal volumes at same $T,p$ contain equal molecules; quantitatively $v \propto n$.

Ideal Gas Equation

• Combine above: $pv = nRT$ (equation of state).
• Universal gas constant $R = 8.314\,\text{J mol}^{-1}\text{K}^{-1}$.

Ideal vs Real Gases

• Ideal: obey $pv=nRT$ always, no intermolecular forces, negligible molecular volume, cannot liquefy (purely theoretical).
• Real: deviate from laws, possess attractions & finite size, can liquefy; approach ideality at high $T$ & low $p$.

Van der Waals Equation (Real Gases)

• Adds corrections: $(p+\frac{a}{v^{2}})(v-b)=RT$ for 1 mol.
  • $a$ (cohesion) corrects pressure; $b$ (co-volume) corrects volume.
• For $n$ mol: $(p+\frac{an^{2}}{v^{2}})(v-nb)=nRT$.
• Critical-point relations (1 mol):
  • $v<em>c = \frac{3RT</em>c}{8p<em>c}$, $b = \frac{v</em>c}{3}$, $a = \frac{27R^{2}T<em>c^{2}}{64p</em>c}$.
  • Compressibility at critical point: $Z<em>c = \frac{p</em>cv<em>c}{RT</em>c}=\frac{3}{8}$.

Compressibility Factor & Corresponding States

• Define $Z=\frac{pv}{RT}=\frac{v<em>{\text{actual}}}{v</em>{\text{ideal}}}$.
  • $Z=1$ ideal; Z>1 positive deviation; Z<1 negative deviation.
• Reduced properties: $p<em>R=\frac{p}{p</em>c},\; T<em>R=\frac{T}{T</em>c},\; v<em>R=\frac{v}{RT</em>c/p_c}$.
• Principle of corresponding states: different gases have nearly same $Z$ at equal $p<em>R,T</em>R$.
• Compressibility chart plots $Z$ vs $p<em>R$ for various $T</em>R$.

Thermodynamic Relations – Ideal Gas

• Specific heats: $C<em>v = \Big(\frac{\partial u}{\partial T}\Big)</em>v$, $C<em>p = \Big(\frac{\partial h}{\partial T}\Big)</em>p$.
• Relation: $C<em>p - C</em>v = R$.
• Changes for mass $m$ between $T<em>1,T</em>2$ (constant $C<em>v,C</em>p$):
  • Internal energy: $\Delta U = mC<em>v\,(T</em>2-T<em>1)$ • Enthalpy: $\Delta H = mC</em>p\,(T<em>2-T</em>1)$
  • Entropy (≡ constant $p$ path): $\Delta S = mC<em>p\,\ln\frac{T</em>2}{T_1}$

Mixtures of Ideal Gases

• Assumptions: each component & mixture behave ideally; no interactions.

Dalton’s Law (pressure)

$P<em>{\text{mix}} = \sum p</em>i$ at common $T,V$.

Amagat’s Law (volume)

$V<em>{\text{mix}} = \sum v</em>i$ at common $T,P$.

Key Definitions

• Mass fraction: $y<em>i = \frac{m</em>i}{\sum m}$; (\sum y_i=1).
• Mole fraction: $x<em>i = \frac{n</em>i}{\sum n}$; (\sum x_i=1).
• Partial pressure: $p<em>i = x</em>i P_{\text{mix}}$.
• Partial volume: $v<em>i = x</em>i V_{\text{mix}}$.
• Molecular weight of mixture: $M = \sum x<em>i M</em>i$.
• Specific gas constant: $R = \frac{\bar R}{M}$ ((\bar R = 8.314\,\text{kJ kmol}^{-1}\text{K}^{-1})).

Specific Heats of Mixture (mass basis)

$C<em>v = \sum y</em>i C<em>{v,i}, \quad C</em>p = \sum y<em>i C</em>{p,i}$.

Gibbs Law (additivity)

Total $U,H,S$ of mixture = sum of components evaluated at common $T,V$.

Quick Reference Equations

• Ideal gas: $pv = RT,\; pV = mRT$.
• Van der Waals: $(p+\frac{a}{v^{2}})(v-b)=RT$.
• Relation: $C<em>p - C</em>v = R$.
• Compressibility: $Z = \frac{pv}{RT}$.
• Dalton: $p<em>i = x</em>i P$; Amagat: $v<em>i = x</em>i V$.