Chemca Formula Sheet - States of Matter

CHEMCA

EXAM MASTER FORMULA SHEET

States of Matter (Gases & Liquids)

Ultimate Revision for JEE Main, Advanced & NEET

Universal Gas Constant (\(R\)) Values:

0.0821 L·atm / K·mol

8.314 J / K·mol

1.987 \(\approx 2\) cal / K·mol

0.0831 L·bar / K·mol

1. Ideal Gas Laws

Boyle's Law (\(T, n\) const)

\(P_1V_1 = P_2V_2\)

\(P \propto 1/V\)

Charles's Law (\(P, n\) const)

\(\frac{V_1}{T_1} = \frac{V_2}{T_2}\)

\(V \propto T\)

Gay-Lussac's (\(V, n\) const)

\(\frac{P_1}{T_1} = \frac{P_2}{T_2}\)

\(P \propto T\)

Ideal Gas Equation:

\[ PV = nRT \implies PV = \frac{w}{M}RT \]

\[ PM = dRT \]

\(M\) = Molar mass, \(d\) = Density of gas

Combined Gas Law:

\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]

Used when a fixed amount of gas undergoes changes in P, V, and T.

2. Dalton's & Graham's Law

Dalton's Law of Partial Pressure:

\[ P_{total} = p_1 + p_2 + p_3 + \dots \text{ (at constant V, T)} \]

Partial Pressure via Mole Fraction: \[ p_i = \chi_i \times P_{total} \]

Gas Collected Over Water: \[ P_{\text{dry gas}} = P_{\text{total}} - \text{Aqueous Tension} \]

Graham's Law of Effusion/Diffusion:

Rate (\(r\)) is inversely proportional to the square root of molar mass or density.

\[ \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} = \sqrt{\frac{d_2}{d_1}} \]

Extended Rate Formulas (\(r\)): \[ \frac{r_1}{r_2} = \frac{V_1/t_1}{V_2/t_2} = \frac{n_1/t_1}{n_2/t_2} = \frac{x_1/t_1}{x_2/t_2} \]

\(V\) = volume effused, \(n\) = moles effused, \(x\) = distance traveled in tube.

3. Kinetic Theory of Gases (KTG) & Speeds

RMS Speed (\(u_{rms}\))

\[ \sqrt{\frac{3RT}{M}} \text{ or } \sqrt{\frac{3P}{d}} \]

Average Speed (\(u_{av}\))

\[ \sqrt{\frac{8RT}{\pi M}} \]

Most Probable (\(u_{mp}\))

\[ \sqrt{\frac{2RT}{M}} \]

Kinetic Energy (\(K.E.\)):

\[ K.E. = \frac{3}{2}nRT \text{ (For n moles)} \]

\[ K.E. = \frac{3}{2}kT \text{ (Per molecule)} \]

\(k = R/N_A = 1.38 \times 10^{-23} \text{ J/K}\). K.E. depends only on absolute temperature.

Maxwell-Boltzmann Distribution:

Speed Ratio: \(u_{mp} : u_{av} : u_{rms} = 1 : 1.128 : 1.224\)
Area under curve = Total number of molecules.
As \(T\) increases, curve flattens and peak shifts to the right (higher speeds, fewer molecules at \(u_{mp}\)).

4. Real Gases & Compressibility

van der Waals Equation (Real Gases):

\[ \left( P + \frac{an^2}{V^2} \right) (V - nb) = nRT \]

Constant 'a' (Attraction):

Measures magnitude of intermolecular attractive forces.

Units: \(\text{atm L}^2 \text{ mol}^{-2}\)

Higher 'a' \(\implies\) easily liquefiable (e.g., \(NH_3 > N_2\)).

Constant 'b' (Volume):

Measures effective size/excluded volume of molecules.

Units: \(\text{L mol}^{-1}\) (\(b = 4 \times N_A \times \frac{4}{3}\pi r^3\))

Compressibility Factor (\(Z\)):

\[ Z = \frac{PV}{nRT} = \frac{V_{real}}{V_{ideal}} \]

\(Z=1\): Ideal Gas
At Low P (\(Z<1\)): Attractive forces dominate. \(Z = 1 - \frac{a}{V_m RT}\)
At High P (\(Z>1\)): Repulsive forces dominate. \(Z = 1 + \frac{Pb}{RT}\)
\(H_2\) and \(He\): \(Z>1\) always at normal temperatures.

Boyle's Temp (\(T_b\)):

Temp where real gas behaves ideally over a range of P.

\[ T_b = \frac{a}{Rb} \]

Critical Constants (\(T_c, P_c, V_c\)):

\[ T_c = \frac{8a}{27Rb} \quad P_c = \frac{a}{27b^2} \quad V_c = 3b \]

\(Z_c = \frac{P_c V_c}{R T_c} = \frac{3}{8} = 0.375\)

5. Liquid State Properties

Vapor Pressure

Pressure exerted by vapor in equilibrium with liquid at constant T.

Clausius-Clapeyron Eq:

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

V.P. \(\uparrow\) as T \(\uparrow\). V.P. \(\downarrow\) as Intermolecular forces \(\uparrow\).

Surface Tension (\(\gamma\))

Force acting per unit length perpendicular to an imaginary line on the surface.

\[ \gamma = \frac{F}{L} \text{ or } \frac{W}{\Delta A} \]

Units: \(N/m\) or \(J/m^2\)

Decreases as Temperature increases.

Viscosity (\(\eta\))

Internal friction between layers of a liquid.

\[ F = \eta A \frac{dv}{dx} \]

Units: Poise (\(g \cdot cm^{-1} \cdot s^{-1}\)) or \(Pa \cdot s\)

Decreases as Temperature increases.

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