Fundamental Laws of Thermodynamics
🔥 First Law of Thermodynamics
$$dU = \delta Q - \delta W$$
$$\Delta U = Q - W$$
Where: U = Internal Energy, Q = Heat added to system, W = Work done by system
♨️ Second Law of Thermodynamics
$$dS \geq \frac{\delta Q}{T}$$
$$\Delta S_{universe} \geq 0$$
Clausius Statement: Heat cannot flow spontaneously from cold to hot.
Kelvin-Planck Statement: No heat engine can convert all heat into work.
❄️ Third Law of Thermodynamics
$$\lim_{T \to 0} S = 0$$
Entropy of a perfect crystal at absolute zero is zero.
⚡ Fundamental Thermodynamic Relations
$$dU = TdS - PdV + \mu dN$$
$$H = U + PV$$
$$F = U - TS$$
$$G = H - TS = U + PV - TS$$
Where: H = Enthalpy, F = Helmholtz Free Energy, G = Gibbs Free Energy
⚠️ Important Work Formulas
$$W = \int P \, dV$$
$$W_{reversible} = nRT \ln\left(\frac{V_f}{V_i}\right)$$ (Isothermal Process)
$$W = \frac{nR(T_f - T_i)}{1-\gamma}$$ (Adiabatic Process)
Thermodynamic Processes
| Process | Condition | ΔU | Heat (Q) | Work (W) |
|---|---|---|---|---|
| Isothermal | \(T = \text{const}\) | 0 | \(nRT\ln\left(\frac{V_f}{V_i}\right)\) | \(nRT\ln\left(\frac{V_f}{V_i}\right)\) |
| Adiabatic | \(Q = 0\) | \(-W\) | 0 | \(\frac{nR(T_f - T_i)}{\ 1 - \gamma\ }\) |
| Isochoric | \(V = \text{const}\) | \(Q\) | \(nC_v(T_f - T_i)\) | 0 |
| Isobaric | \(P = \text{const}\) | \(nC_v(T_f - T_i)\) | \(nC_p(T_f - T_i)\) | \(P(V_f - V_i)\) |
📊 Polytropic Process
$$PV^n = \text{constant}$$
$$W = \frac{P_2V_2 - P_1V_1}{1-n} = \frac{nR(T_2 - T_1)}{1-n}$$
Special Cases:
- n = 0: Isobaric
- n = 1: Isothermal
- n = γ: Adiabatic
- n = ∞: Isochoric
🔄 Reversible vs Irreversible
Reversible Process: \(\Delta S_{universe} = 0\)
Irreversible Process: \(\Delta S_{universe} > 0\)
$$W_{lost} = T_0 \Delta S_{universe}$$
Thermodynamic Variables
| System | Intensive Variable | Extensive Variable |
|---|---|---|
| Hydrostatic system | Pressure, P | Volume, V |
| Stretched wire | Tension, F | Length, L |
| Surface | Surface tension, γ | Area, A |
| Dielectric slab | Electric field, E | Total polarization, P |
| Paramagnetic rod | Magnetic field, H | Total magnetization, M |
Equations of State
| System/Law | Equation |
|---|---|
| Ideal Gas | \(PV = nRT\) |
| van der Waals | \(\left(P + \frac{a}{V^2}\right)(V - b) = nRT\) |
| Stretched Wire | \(F = k(L - L_0)\) |
| Dielectric Slab | \(P = \left(a + \frac{b}{T}\right)E\) |
| Paramagnetic Material (Curie's Law) | \(M = \frac{C}{T}H\) |
First Law of Thermodynamics
| System | First Law Form |
|---|---|
| Hydrostatic system | \(dU = dQ - PdV\) |
| Stretched wire | \(dU = dQ + FdL\) |
| Surface | \(dU = dQ + \gamma dA\) |
| Dielectric slab | \(dU = dQ + EdP\) |
| Paramagnetic rod | \(dU = dQ + \mu_0 H dM\) |
Heat Capacity Definitions
| Term/Concept | Equation |
|---|---|
| Heat Capacity | \(C = \frac{dQ}{dT}\) |
| Specific Heat Capacity | \(c = \frac{1}{m}\frac{dQ}{dT}\) |
| Constant Volume (CV) | \(C_V = \left(\frac{\partial U}{\partial T}\right)_V\) |
| Constant Pressure (CP) | \(C_P = \left(\frac{\partial Q}{\partial T}\right)_P\) |
| Relation between CP & CV | \(C_P - C_V = \left[\left(\frac{\partial U}{\partial V}\right)_T + P\right]V\beta\) |
Thermodynamic Cycles & Heat Engines
🚂 Carnot Cycle
$$\eta_{Carnot} = 1 - \frac{T_C}{T_H} = \frac{T_H - T_C}{T_H}$$
$$\frac{Q_H}{T_H} = \frac{Q_C}{T_C}$$
Steps: Isothermal expansion → Adiabatic expansion → Isothermal compression → Adiabatic compression
⚙️ Otto Cycle
$$\eta_{Otto} = 1 - \frac{1}{r^{\gamma-1}}$$
Where: r = compression ratio = \(V_{max}/V_{min}\)
Steps: Adiabatic compression → Isochoric heating → Adiabatic expansion → Isochoric cooling
🔧 Diesel Cycle
$$\eta_{Diesel} = 1 - \frac{1}{\gamma} \cdot \frac{r_c^{\gamma} - 1}{r^{\gamma-1}(r_c - 1)}$$
Where: r = compression ratio, \(r_c\) = cutoff ratio
❄️ Refrigerators and Heat Pumps
$$COP_{ref} = \frac{Q_C}{W} = \frac{Q_C}{Q_H - Q_C} = \frac{T_C}{T_H - T_C}$$ (Carnot Refrigerator)
$$COP_{HP} = \frac{Q_H}{W} = \frac{Q_H}{Q_H - Q_C} = \frac{T_H}{T_H - T_C}$$ (Carnot Heat Pump)
⚠️ Efficiency Relations
$$\eta = \frac{W_{net}}{Q_{in}} = 1 - \frac{Q_{out}}{Q_{in}}$$
$$\eta_{any} \leq \eta_{Carnot}$$
Entropy & Free Energy
🌀 Entropy Calculations
$$dS = \frac{\delta Q_{rev}}{T}$$
$$S = k_B \ln \Omega$$ (Boltzmann's Formula)
$$\Delta S = nC_v \ln\left(\frac{T_f}{T_i}\right) + nR\ln\left(\frac{V_f}{V_i}\right)$$ (Ideal Gas)
🔄 Entropy Change for Common Processes
| Isothermal: | \(\Delta S = nR\ln\left(\frac{V_f}{V_i}\right) = \frac{Q}{T}\) |
| Isobaric: | \(\Delta S = nC_p\ln\left(\frac{T_f}{T_i}\right)\) |
| Isochoric: | \(\Delta S = nC_v\ln\left(\frac{T_f}{T_i}\right)\) |
| Adiabatic: | \(\Delta S = 0\) (reversible) |
| Phase Change: | \(\Delta S = \frac{L}{T}\) |
⚡ Maxwell Relations
$$\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V$$
$$\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P$$
$$\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V$$
$$\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P$$
🎯 Free Energy Criteria
Helmholtz Free Energy (F): Minimum at equilibrium (constant T, V)
$$dF = -SdT - PdV$$
$$\Delta F = \Delta U - T\Delta S$$
Gibbs Free Energy (G): Minimum at equilibrium (constant T, P)
$$dG = -SdT + VdP$$
$$\Delta G = \Delta H - T\Delta S$$
Kinetic Theory of Gases
🎱 Ideal Gas Relations
$$PV = nRT = Nk_BT$$
$$PV^\gamma = \text{constant}$$ (Adiabatic)
$$\gamma = \frac{C_p}{C_v}$$
📊 Molecular Speeds
$$v_{rms} = \sqrt{\frac{3k_BT}{m}} = \sqrt{\frac{3RT}{M}}$$
$$\bar{v} = \sqrt{\frac{8k_BT}{\pi m}} = \sqrt{\frac{8RT}{\pi M}}$$
$$v_{mp} = \sqrt{\frac{2k_BT}{m}} = \sqrt{\frac{2RT}{M}}$$
Where: \(v_{rms}\) = root mean square speed, \(\bar{v}\) = average speed, \(v_{mp}\) = most probable speed
🌡️ Internal Energy & Heat Capacities
$$U = \frac{f}{2}nRT$$
$$C_v = \frac{f}{2}R$$
$$C_p = C_v + R = \frac{f+2}{2}R$$
$$\gamma = \frac{C_p}{C_v} = \frac{f+2}{f}$$
| Gas Type | f (degrees of freedom) | γ |
|---|---|---|
| Monatomic | 3 | 5/3 ≈ 1.67 |
| Diatomic | 5 | 7/5 = 1.40 |
| Polyatomic | 6 | 4/3 ≈ 1.33 |
📈 Maxwell-Boltzmann Distribution
$$f(v) = 4\pi n \left(\frac{m}{2\pi k_BT}\right)^{3/2} v^2 e^{-mv^2/2k_BT}$$
Energy Distribution:
$$f(E) = \frac{2\pi n}{(\pi k_BT)^{3/2}} \sqrt{E} \, e^{-E/k_BT}$$
🔵 Real Gases - Van der Waals Equation
$$\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$$
Critical Constants:
$$T_c = \frac{8a}{27Rb}, \quad P_c = \frac{a}{27b^2}, \quad V_c = 3nb$$
Constants & Reference Tables
🔢 Fundamental Constants
Gas Constant (R)
8.314 J/(mol·K)
0.08206 L·atm/(mol·K)
Boltzmann Constant (kB)
1.381 × 10-23 J/K
Avogadro's Number (NA)
6.022 × 1023 mol-1
Standard Pressure
1 atm = 101,325 Pa
1 bar = 105 Pa
🔄 Unit Conversions
| Quantity | Conversions |
|---|---|
| Energy | 1 cal = 4.184 J, 1 eV = 1.602 × 10-19 J |
| Temperature | T(K) = T(°C) + 273.15, T(°F) = 9T(°C)/5 + 32 |
| Pressure | 1 Torr = 133.322 Pa, 1 psi = 6,895 Pa |
| Volume | 1 L = 10-3 m³ = 1000 cm³ |
💡 Exam Tips & Common Pitfalls
- Sign Conventions: W is positive when work is done BY the system
- Units: Always check that R has correct units for your calculation
- Reversible Process: Maximum work is obtained in reversible processes
- Entropy: Always increases for isolated systems (2nd Law)
- Carnot Efficiency: Maximum possible efficiency between two temperatures
- State Functions: U, H, S, F, G are path-independent
- Process Functions: Q and W are path-dependent
- Adiabatic: No heat transfer, but temperature can change
- Isothermal: ΔU = 0 for ideal gas (U depends only on T)
📝 Quick Reference Formulas
| Ideal Gas: | PV = nRT |
| First Law: | dU = δQ - δW |
| Carnot Efficiency: | η = 1 - TC/TH |
| Entropy (Boltzmann): | S = kB ln Ω |
| Gibbs Free Energy: | G = H - TS |
| RMS Speed: | vrms = √(3RT/M) |