thermodynamics

# Thermodynamics ## What I do I provide comprehensive expertise in thermodynamics, the branch of physics describing heat, work, and energy transformations. I enable you to apply the laws of thermodynamics, compute entropy and free energy, analyze heat engines and refrigerators, study phase transitions, work with thermodynamic potentials, and apply statistical interpretations. My knowledge spans from macroscopic laws to statistical foundations essential for engineering, chemistry, materials science, and physics. ## When to use me Use thermodynamics when you need to: analyze heat engine efficiency and cycles, calculate equilibrium properties of systems, study phase diagrams and phase transitions, compute chemical reaction spontaneity, design refrigeration and HVAC systems, analyze thermodynamic stability, calculate properties of gases and materials, or apply statistical mechanics to derive macroscopic laws. ## Core Concepts - **Zeroth Law of Thermodynamics**: Thermal equilibrium defines temperature and allows thermometers to measure it. - **First Law of Thermodynamics**: Conservation of energy with heat (Q) and work (W) as energy transfer modes (ΔU = Q - W). - **Second Law of Thermodynamics**: Entropy always increases in isolated systems with heat engines having maximum Carnot efficiency. - **Third Law of Thermodynamics**: Entropy approaches a constant minimum (zero for perfect crystals) as T → 0. - **Entropy and Disorder**: Statistical measure of microscopic configurations consistent with macroscopic state. - **Thermodynamic Potentials**: Internal energy (U), enthalpy (H), Helmholtz free energy (F), and Gibbs free energy (G). - **Phase Transitions**: Changes between states of matter with latent heat and discontinuities in derivatives of free energy. - **Carnot Cycle**: Maximum possible efficiency for heat engines operating between two temperature reservoirs. - **Maxwell Relations**: Derivatives of thermodynamic potentials connected through equality of mixed partials. - **Statistical Interpretation**: Entropy as S = k_B ln(Ω) connecting microscopic and macroscopic descriptions. ## Code Examples ### Thermodynamic Laws and Cycles ```python import numpy as np # First law: ΔU = Q - W def first_law(Q, W): """Calculate internal energy change.""" return Q - W # Work for different processes def work_pressure_volume(P, V1, V2): """Isothermal work: W = ∫PdV = nRT ln(V2/V1)""" return P * V2 - P * V1 def isothermal_work(n, R, T, V1, V2): """Work in isothermal expansion.""" return n * R * T * np.log(V2 / V1) def adiabatic_work(n, R, T1, V1, V2, gamma): """Work in adiabatic expansion.""" return n * R * (T1 - T2) / (gamma - 1) def adiabatic_temperature(T1, V1, V2, gamma): """TV^(γ-1) = constant.""" return T1 * (V1 / V2)**(gamma - 1) # Ideal gas properties R = 8.314 # J/(mol·K) gamma = 1.4 # Air print("First law examples:") n = 1 # mol T1 = 300 # K V1, V2 = 1, 2 # L T2 = adiabatic_temperature(T1, V1, V2, gamma) W_adi = adiabatic_work(n, R, T1, V1, V2, gamma) print(f" Adiabatic expansion V1→2V1 at T={T1}K:") print(f" Final temperature: {T2:.1f} K") print(f" Work done: {W_adi:.1f} J") # Carnot cycle def carnot_efficiency(T_hot, T_cold): """η = 1 - T_cold/T_hot""" return 1 - T_cold / T_hot def carnot_refrigerator_coefficient(T_cold, T_hot): """COP = T_cold/(T_hot - T_cold)""" return T_cold / (T_hot - T_cold) T_hot, T_cold = 500, 300 # K eta_carnot = carnot_efficiency(T_hot, T_cold) cop_carnot = carnot_refrigerator_coefficient(T_cold, T_hot) print(f"\nCarnot cycle:") print(f" T_hot = {T_hot}K, T_cold = {T_cold}K") print(f" Maximum engine efficiency: {eta_carnot*100:.1f}%") print(f" Maximum refrigerator COP: {cop_carnot:.2f}") # Otto cycle (spark ignition engine) def otto_cycle_efficiency(r, gamma): """η = 1 - 1/r^(γ-1)""" return 1 - 1 / (r**(gamma - 1)) def diesel_cycle_efficiency(r, cutoff, gamma): """η = 1 - (1/r^(γ-1)) * ((ρ^γ - 1)/(γ(ρ-1)))""" return 1 - (1 / r**(gamma - 1)) * (cutoff**gamma - 1) / (gamma * (cutoff - 1)) for compression_ratio in [8, 10, 12]: eta = otto_cycle_efficiency(compression_ratio, gamma) print(f" Otto cycle (r={compression_ratio}): η = {eta*100:.1f}%") ``` ### Entropy Calculations ```python import numpy as np from scipy.integrate import quad def entropy_change_isothermal(n, R, V2, V1): """ΔS = nR ln(V2/V1) for isothermal expansion.""" return n * R * np.log(V2 / V1) def entropy_change_temperature(n, Cv, T2, T1): """ΔS = ∫dQ_rev/T = ∫Cv dT/T = Cv ln(T2/T1)""" return n * Cv * np.log(T2 / T1) def entropy_of_mixing(n1, n2, V1, V2): """Entropy of mixing two ideal gases.""" return n1 * R * np.log((V1 + V2) / V1) + n2 * R * np.log((V1 + V2) / V2) # Example: Heating and expanding ideal gas n = 1 # mol Cv = (3/2) * R # Monatomic gas T1, T2 = 300, 600 # K V1, V2 = 1, 2 # L S_heat = entropy_change_temperature(n, Cv, T2, T1) S_expand = entropy_change_isothermal(n, R, V2, V1) print("Entropy calculations:") print(f" Heating 1 mol from 300K to 600K: ΔS = {S_heat:.2f} J/K") print(f" Isothermal expansion V→2V: ΔS = {S_expand:.2f} J/K") print(f" Total ΔS: {S_heat + S_expand:.2f} J/K") # Statistical entropy def boltzmann_entropy(omega): """S = k_B ln(Ω)""" kB = 1.38e-23 return kB * np.log(omega) # Example: 100 coin tosses omega = 2**100 S_coins = boltzmann_entropy(omega) print(f"\nStatistical entropy:") print(f" 100 coin tosses: Ω = {omega:.2e}") print(f" S = {S_coins:.2e} J/K") print(f" S/kB = {np.log(omega):.2f}") # Gibbs paradox def gibbs_paradox(n, V1, V2): """Show entropy of mixing is extensive.""" return 2 * n * R * np.log((V1 + V2) / V1) print(f"\nGibbs paradox:") print(f" Mixing entropy should be proportional to amount of gas") print(f" ΔS = {gibbs_paradox(1, 1, 1):.2f} J/K for 1 mole mixing") # Entropy of water def water_entropy(T, phase='liquid'): """Approximate entropy of water at 298K.""" S_liquid = 69.9 # J/(mol·K) at 298K S_ice = 48.0 # J/(mol·K) at 273K S_vapor = 188.8 # J/(mol·K) at 373K if phase == 'liquid': return S_liquid elif phase == 'solid': return S_ice else: return S_vapor print(f"\nWater entropy at 298K:") print(f" Liquid: {water_entropy(298, 'liquid'):.1f} J/(mol·K)") print(f" Solid: {water_entropy(298, 'solid'):.1f} J/(mol·K)") ``` ### Free Energy and Equilibrium ```python import numpy as np def helmholtz_free_energy(T, V, n, U): """F = U - TS""" return U - T * S def gibbs_free_energy(T, P, n, H): """G = H - TS""" return H - T * S def gibbs_free_energy_reaction(T, dH, dS): """ΔG = ΔH - TΔS""" return dH - T * dS def reaction_spontaneity(dG): """Check if reaction is spontaneous.""" return dG < 0 def equilibrium_constant(dG, T, R=8.314): """K = exp(-ΔG°/RT)""" return np.exp(-dG / (R * T)) # Example: Haber process (N₂ + 3H₂ → 2NH₃) dH_reaction = -92.4e3 # J/mol (exothermic) dS_reaction = -198.7 # J/(mol·K) (decrease in gas moles) T_298 = 298 # K dG_298 = gibbs_free_energy_reaction(T_298, dH_reaction, dS_reaction) K_298 = equilibrium_constant(dG_298, T_298) print("Gibbs free energy (Haber process):") print(f" ΔH° = {dH_reaction/1000:.1f} kJ/mol") print(f" ΔS° = {dS_reaction:.1f} J/(mol·K)") print(f" ΔG°(298K) = {dG_298/1000:.1f} kJ/mol") print(f" K(298K) = {K_equilibrium_constant(dG_298, T_298):.2e}") # Temperature dependence of K for T in [298, 400, 500, 600, 700]: dG = gibbs_free_energy_reaction(T, dH_reaction, dS_reaction) K = equilibrium_constant(dG, T) print(f" K({T}K) = {K:.2e}") # van't Hoff equation def van_t_hoff(K1, K2, T1, T2, dH): """ln(K2/K1) = -ΔH/R (1/T2 - 1/T1)""" return np.log(K2/K1) == -dH/R * (1/T2 - 1/T1) # Maxwell relations def maxwell_relation_dG(P, T, dG_dP, dG_dT): """∂G/∂P = V, ∂G/∂T = -S, then (∂V/∂T)_P = -(∂S/∂P)_T""" pass # Chemical potential def chemical_potential(T, P, mu0, R=8.314): """μ = μ° + RT ln(P/P°)""" return mu0 + R * T * np.log(P / 1e5) # P in Pa def fugacity_coefficient(P, phi): """f = φP for real gases.""" return phi * P print(f"\nChemical potential:") mu0_N2 = -1.5e5 # J/mol at 1 bar for P in [0.1, 1, 10]: # bar mu = chemical_potential(298, P * 1e5, mu0_N2) print(f" μ(N₂, {P} bar) = {mu/1000:.1f} kJ/mol") ``` ### Phase Transitions ```python import numpy as np def clausius_clapeyron(dH, T, dV, R=8.314): """dP/dT = ΔH/(TΔV)""" return dH / (T * dV) def vapor_pressure_clausius(T, P1, T1, dH_vap): """ln(P2/P1) = -ΔH_vap/R (1/T2 - 1/T1)""" return P1 * np.exp(-dH_vap/R * (1/T2 - 1/T1)) def critical_properties(Tc, Pc): """Estimate critical properties.""" Pc_atm = Pc / 1.013e5 # Convert to atm # van der Waals constants a = 27 * R**2 * Tc**2 / (64 * Pc) b = R * Tc / (8 * Pc) return a, b def reduced_properties(T, P, Tc, Pc): """Tr = T/Tc, Pr = P/Pc""" return T / Tc, P / Pc def law_of_rectilinear_diameters(T, T_c, rho_l_c, rho_v_c): """ρ_liq + ρ_vap = 2ρ_c + A(T_c - T)""" A = 0.5 # Typical coefficient return rho_l_c + rho_v_c, 2 * rho_v_c + A * (T_c - T) # Water phase diagram parameters Tc_water = 647.1 # K Pc_water = 22.06e6 # Pa dH_vap_water = 40.7e3 # J/mol print("Phase transition calculations:") print(f" Water critical point: Tc={Tc_water}K, Pc={Pc_water/1e6:.1f} MPa") # Estimate vapor pressure at different temperatures P1 = 101325 # 1 atm T1 = 373.15 # Boiling point at 1 atm dH_vap = 40.7e3 for T in [350, 360, 370, 380, 390]: P_vap = vapor_pressure_clausius(T, P1, T1, dH_vap) print(f" P_vap({T}K) = {P_vap/1000:.1f} kPa") # Clausius-Clapeyron for water at 100°C dH = 40.7e3 # J/mol T = 373.15 # K dV = 30e-3 # m³/mol (approx volume change vapor-liquid) dP_dT = clausius_clapeyron(dH, T, dV) print(f"\nClausius-Clapeyron at 100°C:") print(f" dP/dT = {dP_dT:.2f} Pa/K = {dP_dT/1000:.2f} kPa/K") # Gibbs phase rule def gibbs_phase_rule(C, P): """F = C - P + 2 for non-reactive systems.""" return C - P + 2 print(f"\nGibbs phase rule:")