Review

Checklist of Key Concepts¶

Section 1.1. Course Introduction¶

• Thermodynamic Systems

  • System vs. Surroundings

  • Types: isolated (no exchange of energy or matter), closed (exchanges energy but not matter), open (exchanges energy and matter)

• State Variables and Equilibrium

  • State variables (e.g., $P$, $V$, $T$) and other state functions (e.g., internal energy $U$)

  • Path functions (e.g., heat $q$, work $w$) depend on the path taken between states

  • Equilibrium implies no net change in macroscopic properties over time

• Energy, Heat, and Work

  • Kinetic vs. Potential energy

  • Heat: energy transfer due to temperature difference

  • Work: energy transfer when a force acts through a distance

Section 1.2. Kinetic Theory¶

• Kinetic Theory Assumptions

  1. Large number of particles (identical, random motion)

  2. Negligible particle volume compared to container volume

  3. Collisions are elastic (no energy loss)

  4. No long-range interparticle forces (except during collisions)

  5. Newton’s laws (classical mechanics) apply

• Pressure Origin

  • Microscopic collisions of particles with container walls

  • Force per unit area results from changes in particle momentum

• Equipartition Theorem

  • Average translational kinetic energy per particle $\langle E_{\mathrm{kin}} \rangle = \frac{3}{2} k_{\mathrm{B}} T$

  • Relates $T$ to the mean-square speed

Section 1.3. Ideal Gases¶

• Classical Gas Laws

  • Boyle ($P \propto 1/V$ at constant $T,\,N$)

  • Charles ($V \propto T$ at constant $P,\,N$)

  • Gay-Lussac ($P \propto T$ at constant $V,\,N$)

  • Avogadro ($V \propto N$ at constant $P,\,T$)

• Ideal Gas Equation

  • $PV = N k_{\mathrm{B}} T = n R T$

  • Assumes point-like, non-interacting particles

• Absolute Temperature Scale

  • Kelvin scale derived by extrapolating volumes to zero at $-273.15\,\text{°C}$

Section 1.4. Real Gases¶

• Deviations from Ideal Behavior

  • Become significant at high $P$ (high density) or low $T$

  • Compressibility factor $Z = \frac{PV}{nRT}$ indicates deviation ($Z=1$ ideal; $Z<1$ net attraction; $Z>1$ net repulsion)

• van der Waals Equation

  • Introduces parameters $a$ (accounts for attractions) and $b$ (excluded volume)

  • Shows the characteristic van der Waals “loop” below the critical temperature (associated with liquid–vapor coexistence)

• Critical Point and Corresponding States

  • Critical conditions $(T_c, P_c, V_{m,c})$ mark the end of the liquid–vapor coexistence line

  • Reduced variables $T_r = T/T_c,\; P_r = P/P_c,\; V_{m,r} = V_m / V_{m,c}$ collapse different substances onto similar curves (principle of corresponding states)

Checklist of Most Important Equations¶

1. Ideal Gas Law

   $$PV = N k_{\mathrm{B}} T = n R T.$$

   (1)

   • Applicability: low pressures, relatively high temperatures, or low densities (particles effectively non-interacting).

2. Pressure from Kinetic Theory

   $$P = \frac{N m \langle v^2 \rangle}{3V}.$$

   (2)

   • Applicability: idealized gas of point particles with elastic collisions; derived under kinetic-theory assumptions.

3. Average Kinetic Energy / Equipartition

   $$\langle E_{\mathrm{kin}} \rangle \;=\; \frac{3}{2} k_{\mathrm{B}} T \quad\Longleftrightarrow\quad \frac{1}{2} m \langle v^2 \rangle = \frac{3}{2} k_{\mathrm{B}} T.$$

   (3)

   • Applicability: classical (high-temperature) regime, ignoring quantum effects and molecular internal modes.

4. Root-Mean-Square (rms) Speed

   $$v_{\mathrm{rms}} = \sqrt{\frac{3 k_{\mathrm{B}} T}{m}}.$$

   (4)

   • Applicability: same assumptions as equipartition (kinetic theory of gases).

5. Compressibility Factor

   $$Z = \frac{PV}{nRT}.$$

   (5)

   • Interpretation:

     • $Z = 1$: ideal gas

     • $Z < 1$: net attractive forces

     • $Z > 1$: net repulsive forces

   • Applicability: any real gas to quantify deviation from ideality.

6. van der Waals Equation of State (in molar form)

   $$\left(P + \frac{a_m}{V_m^2}\right)\;\left(V_m - b_m\right)\;=\;R\,T.$$

   (6)

   • Applicability: real gases at moderate deviations from ideality; fails under extreme conditions (very high $P$, near liquefaction, etc.).

7. Critical-Point Relationships (van der Waals)

   $$V_{m,c} = 3\,b_m,\quad P_c = \frac{a_m}{27\,b_m^2},\quad T_c = \frac{8\,a_m}{27\,b_m\,R}.$$

   (7)

   • Interpretation: defines the critical temperature $T_c$, critical pressure $P_c$, and critical molar volume $V_{m,c}$ for a van der Waals fluid.

8. Corresponding States (reduced variables)

   $$\left(P_r + \frac{3}{V_{m,r}^2}\right)\;\left(V_{m,r} - \frac{1}{3}\right)\;=\;\frac{8}{3}\,T_r \quad \text{where} \quad P_r=\frac{P}{P_c},\quad T_r=\frac{T}{T_c},\quad V_{m,r}=\frac{V_m}{V_{m,c}}.$$

   (8)

   • Applicability: near or above critical conditions for fluids that approximate van der Waals behavior; demonstrates “universal” behavior across substances when scaled by critical properties.