First law: ΔU = Q − W (Q = heat added to system, W = work done by system). Work: isobaric W = PΔV; isochoric W = 0; isothermal W = nRT ln(V_f/V_i); adiabatic W = (P_i V_i − P_f V_f)/(γ−1). Specific heats: C_p − C_v = R; U = n C_v T. Second law: ΔS_universe ≥ 0; ΔS = Q_rev/T. Kinetic theory: v_rms = √(3RT/M), P = (1/3)ρ v_rms².
Work in thermodynamic process (ideal gas)
Process type
P (Pa), ΔV (m³) or V_i, V_f (m³), n (mol), T (K), γ
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First law & work formulas
| Concept | Formula |
|---|---|
| First law | ΔU = Q − W |
| Isobaric | W = P ΔV |
| Isochoric | W = 0 |
| Isothermal | W = nRT ln(V_f/V_i) = nRT ln(P_i/P_f) |
| Adiabatic | W = (P_i V_i − P_f V_f)/(γ−1) = n C_v (T_i − T_f) |
Specific heats & degrees of freedom
| Gas | f | C_v (molar) | C_p (molar) | γ |
|---|---|---|---|---|
| Monatomic | 3 | (3/2)R | (5/2)R | 5/3 ≈ 1.667 |
| Diatomic (room) | 5 | (5/2)R | (7/2)R | 7/5 = 1.4 |
| Diatomic (vibration) | 7 | (7/2)R | (9/2)R | 9/7 ≈ 1.286 |
| Polyatomic (non-linear) | 6 | 3R | 4R | 4/3 ≈ 1.333 |
Mayer: C_p − C_v = R. Internal energy: U = n C_v T (ideal gas).
Internal energy U = n C_v T
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Ideal gas processes summary
| Process | ΔU | Q | W | P–V / T |
|---|---|---|---|---|
| Isothermal | 0 | W | nRT ln(V_f/V_i) | PV = const, T = const |
| Adiabatic | n C_v ΔT | 0 | −ΔU | PV^γ = const, TV^(γ−1) = const |
| Isochoric | n C_v ΔT | ΔU | 0 | V = const, P ∝ T |
| Isobaric | n C_v ΔT | n C_p ΔT | P ΔV | P = const, V ∝ T |
Second law & entropy
| Statement / Concept | Expression |
|---|---|
| Kelvin–Planck | No engine converts heat fully to work |
| Clausius | Heat cannot flow cold→hot without work |
| Entropy change (reversible) | ΔS = Q_rev / T |
| Total entropy | ΔS_universe ≥ 0 |
| Ideal gas | ΔS = n C_v ln(T_f/T_i) + n R ln(V_f/V_i) |
Entropy change (ideal gas)
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Kinetic theory formulas
| Concept | Formula |
|---|---|
| Pressure | P = (1/3) ρ v_rms² |
| v_rms | √(3RT / M) |
| v_avg | √(8RT / (πM)) |
| v_mp | √(2RT / M) |
| Mean free path | λ = 1 / (√2 π d² N/V) |
| Internal energy | U = (f/2) n R T |
Speeds & pressure (ideal gas)
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About this page
For more on temperature scales, Zeroth law, sensible/latent heat, heat engine efficiency, and refrigerator COP, see the Thermal Energy tool. Here we cover first law and work in processes, specific heats and degrees of freedom, process summary, second law and entropy, and kinetic theory of gases.