State Function: A thermodynamic property whose value depends only on the current state of the system (initial and final states) and is independent of the path taken to reach that state
.Path Function: A thermodynamic property whose value depends on the specific path or process followed during the change between states
.The Zeroth Law of Thermodynamics states that if two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other.
Explanation: If system A is in thermal equilibrium with system C, and system B is also in thermal equilibrium with system C, then A and B are at the same temperature. This law provides the basis for the measurement of temperature using thermometers.
Exact Differential: A differential of a state function (like dU or dH) where the integral depends only on the initial and final points. Mathematically, it satisfies Euler's reciprocity relation.
Inexact Differential: A differential of a path function (like dw or dq) where the integral depends on the path taken. These are often denoted by a cross or delta symbol (δw, δq).
| Isothermal Process | Adiabatic Process |
|---|---|
| Occurs at a constant temperature (ΔT = 0). | Occurs with no heat exchange with surroundings (q = 0). |
| System is in thermal contact with a heat reservoir. | System is thermally insulated from the surroundings. |
For an adiabatic process, dq = 0. From First Law: dU = dw = -P dV
.For 1 mole of an ideal gas, dU = Cv dT and P = RT/V
.Cv dT = -(RT/V) dV
(Cv/R) (dT/T) = -(dV/V)
Integrating both sides between (T1, V1) and (T2, V2):
(Cv/R) ln(T2/T1) = -ln(V2/V1) = ln(V1/V2)
Since R = Cp - Cv, then R/Cv = (Cp/Cv) - 1 = γ - 1
.T V^(γ-1) = constant
Work done in irreversible expansion against constant external pressure (Pext):
w = -Pext (V2 - V1)
w = -1 atm * (30 dm³ - 10 dm³) = -1 atm * 20 dm³
To convert to Joules (1 L-atm = 101.325 J):
w = -20 * 101.325 J = -2026.5 J
In a reversible isothermal process, T is constant and P = nRT/V
.w = -∫ P dV (from V1 to V2)
w = -∫ (nRT/V) dV = -nRT ∫ (1/V) dV
w = -nRT [ln V] (from V1 to V2)
w = -nRT ln(V2/V1) or w = -2.303 nRT log(V2/V1)
Combustion of Glucose: C6H12O6(s) + 6O2(g) → 6CO2(g) + 6H2O(l)
ΔH_comb = [6 * ΔHf(CO2) + 6 * ΔHf(H2O)] - [ΔHf(C6H12O6) + 6 * ΔHf(O2)]
-2816 = [6 * (-393.5) + 6 * (-285.9)] - [ΔHf(C6H12O6) + 0]
-2816 = [-2361 - 1715.4] - ΔHf(C6H12O6)
-2816 = -4076.4 - ΔHf(C6H12O6)
ΔHf(C6H12O6) = -4076.4 + 2816
ΔHf(C6H12O6) = -1260.4 kJ/mol
Kirchhoff's equation describes the variation of enthalpy of reaction with temperature
.ΔH = H(products) - H(reactants)
Differentiating with respect to T at constant pressure:
(d(ΔH)/dT)p = (dH(prod)/dT)p - (dH(react)/dT)p
Since (dH/dT)p = Cp:
(d(ΔH)/dT)p = ΔCp
Integrating between T1 and T2:
ΔH2 - ΔH1 = ΔCp (T2 - T1)
The entropy of the universe (an isolated system) increases in the course of any spontaneous change.
Mathematically, for a spontaneous process: ΔS_total > 0
.We know G = H - TS. For a process: ΔG = ΔH - TΔS.
From Maxwell relations, (dΔG/dT)p = -ΔS
.Substituting -ΔS in the ΔG equation:
ΔG = ΔH + T(dΔG/dT)p
Rearranging this gives the Gibbs-Helmholtz equation:
[d(ΔG/T) / dT]p = -ΔH / T²
Work function A is defined as A = U - TS. At constant temperature: ΔA = ΔU - TΔS.
From Second Law, for a reversible process: ΔS = q_rev / T, so TΔS = q_rev
.ΔA = ΔU - q_rev. From First Law: ΔU - q = w.
Therefore, ΔA = w_rev
.Since work done by the system is -w, then -ΔA = -w_rev = w_max.
The decrease in work function (Helmholtz free energy) at constant temperature equals the maximum work obtainable from the system.
The triple point is the specific temperature and pressure at which all three phases (solid, liquid, and gas) of a substance coexist in thermodynamic equilibrium
. For water, the triple point is at 273.16 K and 0.006 atm.The phase diagram of CO2 consists of three areas representing Solid, Liquid, and Gas phases
.If a system at equilibrium is subjected to a change in concentration, temperature, or pressure, the system will shift its equilibrium position in a direction that tends to counteract the effect of the change.
Example: In the Haber process (exothermic), increasing pressure shifts equilibrium toward ammonia production to reduce the number of moles
.The total Gibbs free energy of a mixture is G = Σ ni μi
.Differentiating G: dG = Σ ni dμi + Σ μi dni
.Also, from the fundamental equation at constant T and P: dG = Σ μi dni
.Comparing the two equations:
Σ ni dμi = 0
This shows that chemical potentials of components in a mixture are not independent
.pH Definition: pH is the negative logarithm (base 10) of the molar concentration of hydrogen ions: pH = -log[H+]
.Calculation: Since the concentration is very dilute (10⁻⁹ M), we must include [H+] from water (10⁻⁷ M)
.[H+]_total = [H+]_acid + [H+]_water ≈ 1.01 * 10⁻⁷ M.
pH = -log(1.01 * 10⁻⁷) ≈ 6.99
.