💨 Gaseous State & Real Gases

The gaseous state provides the clearest bridge between microscopic molecular dynamics and macroscopic thermodynamic state variables ($P,V,T$). Advanced problems hinge on Maxwell speed distributions, effusion under non-steady states, Van der Waals compressibility curves ($Z$ vs $P$), and liquefaction critical parameters.

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1. 🎈 Ideal Gas Dynamics & Kinetic Theory

1.1 Derivation of Kinetic Gas Equation

From momentum exchange of $N$ molecules of mass $m$ in a cubic container of volume $V$:

$$P=\frac{1}{3}\frac{mN}{V}{v}_{\text{rms}}^2=\frac{1}{3}\rho v_{\text{rms}}^2$$

Total Translational Kinetic Energy of $n$ moles of an ideal gas:

$$E_K=\frac{3}{2}nRT=\frac{3}{2}N{k}_{B}T$$

Average Kinetic Energy per molecule:

$$\overline{ϵ}=\frac{3}{2}{k}_{B}T(\text{strictly a function of }T\text{ only})$$

1.2 The Three Characteristic Molecular Speeds

$$\text{RMS Speed: }{v}_{\text{rms}}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3k_{B}T}{m}}=1.732\sqrt{\frac{RT}{M}}$$$$\text{Average Speed: }{v}_{\text{avg}}=\sqrt{\frac{8RT}{\pi M}}=1.596\sqrt{\frac{RT}{M}}$$$$\text{Most Probable Speed: }{v}_{\text{mp}}=\sqrt{\frac{2RT}{M}}=1.414\sqrt{\frac{RT}{M}}$$$${\mathbf{v}}_{\text{rms}}\mathbf{>}{\mathbf{v}}_{\text{avg}}\mathbf{>}{\mathbf{v}}_{\text{mp}}(\text{Ratio }\sqrt{3}:\sqrt{8/\pi }:\sqrt{2}\approx 1.224:1.128:1.000)$$

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2. 📈 Maxwell-Boltzmann Distribution of Molecular Speeds

The fraction of molecules having speeds between $v$ and $v+dv$:

$$\frac{d{N}_v}{N}=4\pi {\left(\frac{M}{2\pi RT}\right)}^{3/2}{v}^2\exp (-\frac{M{v}^2}{2RT})dv$$

Key Geometric Features of the Distribution Curve:

1. The peak of the curve represents the most probable speed ($v_{\text{mp}}$).
2. As temperature increases ($T_2>T_1$):
   • The curve shifts to the right (higher speed).
   • The peak height decreases (broadening of the distribution), because total area under the curve is normalized to $1$.
3. Heavier gases (larger $M$) at the same temperature have sharper, narrower distributions shifted to the left.

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3. 🛑 Real Gases & The Van der Waals Equation

Ideal gas theory assumes:

1. Zero volume occupied by gas molecules.
2. Zero intermolecular forces of attraction.

Both assumptions fail at High Pressure and Low Temperature.

$$(P+\frac{a{n}^2}{V^2})(V-nb)=nRT$$

For $1\text{ mole}$ ($n=1,V_m=V/n$):

$$(P+\frac{a}{V_m^2})(V_m-b)=RT$$

3.1 Physical Meaning of Constants $a$ and $b$

• Van der Waals constant $a$ ($[a]={\text{atm L}}^2{\text{mol}}^{-2}={\text{N m}}^4{\text{mol}}^{-2}$):
  • Measures the magnitude of intermolecular attractive forces.
  • Greater value of $a⟹$ gas is more easily liquefied ($S{O}_2>N{H}_3>C{O}_2>C{H}_4>N_2>H_2>He$).
• Van der Waals constant $b$ ($[b]={\text{L mol}}^{-1}={\text{m}}^3{\text{mol}}^{-1}$):
  • Effective excluded volume (co-volume) per mole of molecules:$$b=4N_A\times v_{\text{molecule}}=4N_A\left(\frac{4}{3}\pi r^3\right)=16\pi N_A{r}^3$$
  • The excluded volume is 4 times the actual hard-sphere volume of the molecules!

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4. 📊 Compressibility Factor ($Z$) Analysis

$$Z=\frac{P{V}_m}{RT}=\frac{V_{\text{real}}}{V_{\text{ideal}}}$$

Pressure Regime	Dominant Factor	Simplified Van der Waals Form	Compressibility Factor $Z$	Slope on $Z-P$ Graph
Very Low Pressure ($P\to 0$)	Ideal behavior	$P{V}_m=RT$	$Z=1$	$0$
Low to Moderate Pressure	Intermolecular attractions dominate	$(P+\frac{a}{V_m^2})V_m=RT$	$Z=1-\frac{a}{V_{m}RT}\approx 1-\frac{aP}{(RT{)}^2}$	Negative slope ($Z<1$, gas more compressible)
High Pressure ($P\gg 1\text{ atm}$)	Molecular size / repulsion dominates	$P(V_m-b)=RT$	$Z=1+\frac{Pb}{RT}$	Positive slope ($Z>1$, gas less compressible)
$H_2$ and $He$ at $298\text{ K}$	Extremely weak attraction ($a\approx 0$)	$P(V_m-b)=RT$	$Z=1+\frac{Pb}{RT}>1$ at all $P$	Always positive slope ($Z>1$)

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5. 🧊 Critical Phenomena & Inversion

At the critical point $(T_c,P_c,V_c)$, the $P-V$ isotherm has a point of inflection:

$${\left(\frac{\partial P}{\partial V}\right)}_{T_c}=0\text{and}{\left(\frac{{\partial }^{2}P}{\partial V^2}\right)}_{T_c}=0$$$$V_c=3b,P_c=\frac{a}{27b^2},T_c=\frac{8a}{27Rb}$$$$\text{Critical Compressibility Factor: }{Z}_c=\frac{P_c{V}_c}{R{T}_c}=\frac{3}{8}=0.375(\text{universal for all Van der Waals gases})$$

• Boyle Temperature ($T_B$): Temperature at which a real gas obeys ideal gas laws over an appreciable pressure range ($Z\approx 1$):$$T_B=\frac{a}{Rb}$$
• Inversion Temperature ($T_i$): Temperature above which a gas warms upon Joule-Thomson expansion and below which it cools:$$T_i=2T_B=\frac{2a}{Rb}$$