Review#
Checklist of Key Concepts#
Section 1.1. Course Introduction#
Thermodynamic Systems
System vs. Surroundings
Types: isolated (no exchange), closed (energy exchange only), open (energy + matter exchange)
State Variables and Equilibrium
State variables (e.g., \(P\), \(V\), \(T\)) vs. state functions (e.g., enthalpy \(H\), internal energy \(U\))
Path functions (e.g., heat \(q\), work \(w\)) depend on how a process is carried out
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
Large number of particles (identical, random motion)
Negligible particle volume compared to container volume
Collisions are elastic (no energy loss)
No long-range interparticle forces (except during collisions)
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 \(\langle E_{\text{kin}} \rangle = \frac{3}{2} k_{\text{B}} T\)
Relates temperature to average particle velocity
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
\(PV = N k_{\text{B}} T = n R T\)
Assumes point-like, non-interacting particles
Absolute Temperature Scale
Kelvin scale derived by extrapolating volumes to zero at \(-273.15\,^\circ\text{C}\)
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 characteristic “loop” (liquid–vapor coexistence) below critical temperature
Critical Point and Corresponding States
Critical conditions \((T_c, P_c, V_{\text{m},c})\) mark the end of the liquid–vapor coexistence line
Reduced variables \(T_r = T/T_c,\; P_r = P/P_c,\; V_{m,r} = V_m / V_{\text{m},c}\) collapse different substances onto similar curves (principle of corresponding states)
Checklist of Most Important Equations#
Ideal Gas Law
\[ PV = N k_{\text{B}} T = n R T. \]Applicability: low pressures, relatively high temperatures, or low densities (particles effectively non-interacting).
Pressure from Kinetic Theory
\[ P = \frac{N m \langle v^2 \rangle}{3V}. \]Applicability: idealized gas of point particles with elastic collisions; derived under kinetic-theory assumptions.
Average Kinetic Energy / Equipartition
\[ \langle E_{\text{kin}} \rangle \;=\; \frac{3}{2} k_{\text{B}} T \quad\Longleftrightarrow\quad \frac{1}{2} m \langle v^2 \rangle = \frac{3}{2} k_{\text{B}} T. \]Applicability: classical (high-temperature) regime, ignoring quantum effects and molecular internal modes.
Root-Mean-Square (rms) Speed
\[ v_{\text{rms}} = \sqrt{\frac{3 k_{\text{B}} T}{m}}. \]Applicability: same assumptions as equipartition (kinetic theory of gases).
Compressibility Factor
\[ Z = \frac{PV}{nRT}. \]Interpretation:
\(Z = 1\): ideal gas
\(Z < 1\): net attractive forces
\(Z > 1\): net repulsive forces
Applicability: any real gas to quantify deviation from ideality.
van der Waals Equation of State (in molar form)
\[ \left(P + \frac{a_m}{V_m^2}\right)\;\left(V_m - b_m\right)\;=\;R\,T. \]Applicability: real gases at moderate deviations from ideality; fails under extreme conditions (very high \(P\), near liquefaction, etc.).
Critical-Point Relationships (van der Waals)
\[ V_{\text{m},c} = 3\,b_m,\quad P_c = \frac{a_m}{27\,b_m^2},\quad T_c = \frac{8\,a_m}{27\,b_m\,R}. \]Interpretation: defines the critical temperature \(T_c\), critical pressure \(P_c\), and critical molar volume \(V_{\text{m},c}\) for a van der Waals fluid.
Corresponding States (reduced variables)
\[ \left(P_r + \frac{3}{V_{m,r}^2}\right)\;\left(V_{m,r} - \frac{1}{3}\right)\;=\;\frac{8}{3}\,T_r \quad \text{where} \quad P_r=\frac{P}{P_c},\quad T_r=\frac{T}{T_c},\quad V_{m,r}=\frac{V_m}{V_{\text{m},c}}. \]Applicability: near or above critical conditions for fluids that approximate van der Waals behavior; demonstrates “universal” behavior across substances when scaled by critical properties.