Help me with formula in physical chemistry advance level

here are the key formulas in Physical Chemistry for advanced level:

1. Thermodynamics

• First Law of Thermodynamics: $$\Delta U = Q - W$$ (where $W$ is work done by the system)
• Enthalpy Change: $$\Delta H = \Delta U + P\Delta V$$ $$\Delta H = Q_p \quad \text{(at constant pressure)}$$
• Entropy Change: $$\Delta S = \frac{Q_{rev}}{T}$$ $$\Delta S_{total} = \Delta S_{sys} + \Delta S_{surr} \ge 0$$
• Gibbs Free Energy Change: $$\Delta G = \Delta H - T\Delta S$$ $$\Delta G = -RT \ln K \quad \text{(at equilibrium)}$$ $$\Delta G = \Delta G^\circ + RT \ln Q$$

2. Chemical Kinetics

• Rate Law: $$\text{Rate} = k[A]^m[B]^n$$
• Integrated Rate Laws:
  • First Order: $$\ln[A]_t = \ln[A]_0 - kt$$ $$t_{1/2} = \frac{\ln 2}{k}$$
  • Second Order: $$\frac{1}{[A]_t} = \frac{1}{[A]_0} + kt$$ $$t_{1/2} = \frac{1}{k[A]_0}$$
• Arrhenius Equation: $$k = A e^{-E_a/RT}$$ $$\ln k = \ln A - \frac{E_a}{RT}$$

3. Chemical Equilibrium

• Equilibrium Constant: $$K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}$$ $$K_p = K_c(RT)^{\Delta n}$$
• Van't Hoff Equation: $$\ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H^\circ}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$$

4. Electrochemistry

• Nernst Equation: $$E = E^\circ - \frac{RT}{nF}\ln Q$$
• Relationship between $\Delta G^\circ$ and $E^\circ$: $$\Delta G^\circ = -nFE^\circ$$
• Faraday's Law of Electrolysis: $$Q = It = nF$$ (where $n$ is moles of electrons, $F$ is Faraday constant)

5. Quantum Chemistry (Basic)

• Planck's Equation: $$E = hf = \frac{hc}{\lambda}$$
• De Broglie Wavelength: $$\lambda = \frac{h}{p} = \frac{h}{mv}$$