JEE Special

1. Atomic Structure (Quantum Mechanical Model)

De Broglie Wavelength:

$\lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{\sqrt{2m(K.E.)}}$

Heisenberg’s Uncertainty Principle:

$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$

Radius of $n^{th}$ Bohr Orbit:

$r_n = 0.529 \times \frac{n^2}{Z} \text{ Å}$

Velocity of Electron in $n^{th}$ Orbit:

$v_n = 2.18 \times 10^6 \times \frac{Z}{n} \text{ m/s}$

---

2. Gaseous State (Real Gases)

Van der Waals Equation (for 1 mole):

$\left( P + \frac{a}{V^2} \right)(V - b) = RT$

Critical Constants:

• Critical Temp: $T_c = \frac{8a}{27Rb}$
• Critical Pressure: $P_c = \frac{a}{27b^2}$
• Critical Volume: $V_c = 3b$

Root Mean Square Velocity ($u_{rms}$):

$u_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3P}{d}}$

---

3. Thermodynamics (JEE Advanced Level)

Entropy Change ($\Delta S$):

$\Delta S = nC_v \ln \frac{T_2}{T_1} + nR \ln \frac{V_2}{V_1}$

Joule-Thomson Coefficient ($\mu_{JT}$):

$\mu_{JT} = \left( \frac{\partial T}{\partial P} \right)_H$

Clapeyron-Clausius Equation:

$\log \frac{P_2}{P_1} = \frac{\Delta H_{vap}}{2.303 R} \left[ \frac{T_2 - T_1}{T_1 T_2} \right]$

---

4. Chemical Equilibrium

Relation between $K_p$ and $K_c$:

$K_p = K_c(RT)^{\Delta n_g}$

Reaction Quotient ($Q$) and $\Delta G$:

$\Delta G = \Delta G^\circ + RT \ln Q$

Van’t Hoff Isotherm:

$\Delta G^\circ = -RT \ln K_{eq}$

---

5. Coordination Chemistry (Crystal Field Theory)

Crystal Field Splitting Energy ($\Delta_o$):

$\text{CFSE} = [-0.4n_{t2g} + 0.6n_{eg}]\Delta_o + nP$

Magnetic Moment (Spin only):

$\mu = \sqrt{n(n+2)} \text{ B.M.}$

[Image of d-orbital splitting in octahedral and tetrahedral crystal fields]

---

6. Electrochemistry (Advanced)

Relation between $E_{cell}$ and $K_{eq}$:

$\log K_{eq} = \frac{n E^\circ_{cell}}{0.0591}$

Faraday's First Law of Electrolysis:

$W = ZIt = \frac{E \cdot I \cdot t}{96500}$

---

7. Liquid Solutions

Van’t Hoff Factor ($i$):

$i = 1 + (n-1)\alpha \quad \text{(for dissociation)}$

$i = 1 + (\frac{1}{n}-1)\alpha \quad \text{(for association)}$