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States of Matter Formulas for JEE 2027
The States of Matter chapter is important for building a strong foundation in Chemistry for 2027 JEE aspirants. This chapter mainly covers gas laws, the ideal gas equation, kinetic molecular theory, real gases, and liquid properties. Concepts such as pressure, volume, temperature, and molecular speeds help students understand the behaviour of gases and liquids, strengthening their conceptual understanding for JEE Main and JEE Advanced preparation.
The chapter also includes deviations from ideal gas behaviour, the van der Waals equation, and properties such as viscosity and surface tension. Understanding these concepts and practising numerical problems can help students apply formulas correctly and avoid calculation errors. For quick revision, students can refer to a well-organised JEE Mains Chemistry Formula PDF 2027 to review important equations, definitions, and relationships in one place.
Gas Laws Formulas
Boyle's Law
At constant $$T$$ and $$n$$:
$$$PV = \text{constant} \quad \text{or} \quad P_1 V_1 = P_2 V_2$$$
Worked Example: Boyle's Law
A gas occupies 500 mL at 1 atm. What is its volume at 2.5 atm (temperature unchanged)?
$$P_1 V_1 = P_2 V_2 \implies V_2 = \dfrac{1 \times 500}{2.5} = \textbf{200 mL}$$
Charles's Law
At constant $$P$$ and $$n$$:
$$$\frac{V}{T} = \text{constant} \quad \text{or} \quad \frac{V_1}{T_1} = \frac{V_2}{T_2}$$$
Important: Temperature must be in Kelvin ($$T_K = T_C + 273.15$$).
Gay-Lussac's Law
At constant $$V$$ and $$n$$:
$$$\frac{P}{T} = \text{constant} \quad \text{or} \quad \frac{P_1}{T_1} = \frac{P_2}{T_2}$$$
Avogadro's Law
At constant $$T$$ and $$P$$:
$$$\frac{V}{n} = \text{constant} \quad \text{or} \quad \frac{V_1}{n_1} = \frac{V_2}{n_2}$$$
At STP ($$0°$$C, 1 atm), 1 mole of any ideal gas occupies 22.4 L.
Ideal Gas Equation Formulas
Ideal Gas Equation
$$$PV = nRT$$$
where $$P$$ = pressure, $$V$$ = volume, $$n$$ = moles, $$R$$ = gas constant, $$T$$ = temperature (K).
Values of the Gas Constant $$R$$
| Value | Units | When to use |
|---|---|---|
| $$0.0821$$ | L·atm·mol$$^{-1}$$·K$$^{-1}$$ | $$P$$ in atm, $$V$$ in L |
| $$8.314$$ | J·mol$$^{-1}$$·K$$^{-1}$$ | SI units ($$P$$ in Pa, $$V$$ in m$$^3$$) |
| $$2$$ | cal·mol$$^{-1}$$·K$$^{-1}$$ | Energy in calories (approx.) |
| $$0.0831$$ | L·bar·mol$$^{-1}$$·K$$^{-1}$$ | $$P$$ in bar, $$V$$ in L |
Worked Example: Ideal Gas Equation
Find the volume of 2 moles of an ideal gas at $$27°$$C and 1 atm.
$$V = \dfrac{nRT}{P} = \dfrac{2 \times 0.0821 \times 300}{1} = \textbf{49.26 L}$$
Gas Density and Molar Mass
$$$PM = dRT \quad \text{or} \quad d = \frac{PM}{RT}$$$
where $$d = \dfrac{m}{V}$$ is the density and $$M$$ is the molar mass.
Rearranging: $$M = \dfrac{dRT}{P}$$
Worked Example: Molar Mass from Density
A gas has a density of 1.96 g/L at STP. Find its molar mass.
$$M = \dfrac{dRT}{P} = \dfrac{1.96 \times 0.0821 \times 273}{1} = 43.9 \; \text{g/mol}$$
This is close to 44 g/mol, suggesting the gas is CO$$_2$$.
Dalton's Law and Graham's Law Formulas
Dalton's Law of Partial Pressures
$$$P_{\text{total}} = p_1 + p_2 + p_3 + \cdots$$$
The partial pressure of gas $$i$$ is:
$$$p_i = x_i \times P_{\text{total}}$$$
where $$x_i = \dfrac{n_i}{n_{\text{total}}}$$ is the mole fraction of gas $$i$$.
Worked Example: Partial Pressures
A 10 L container holds 2 mol N$$_2$$ and 3 mol O$$_2$$ at 300 K. Find the total pressure and partial pressures.
$$P_{\text{total}} = \dfrac{n_{\text{total}} RT}{V} = \dfrac{5 \times 0.0821 \times 300}{10} = 12.315 \; \text{atm}$$
$$p_{\text{N}_2} = 0.4 \times 12.315 = 4.926 \; \text{atm}$$, $$p_{\text{O}_2} = 0.6 \times 12.315 = 7.389 \; \text{atm}$$
Graham's Law of Diffusion
At constant $$T$$ and $$P$$:
$$$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} = \sqrt{\frac{d_2}{d_1}}$$$
where $$r$$ = rate of diffusion/effusion, $$M$$ = molar mass, $$d$$ = density. Lighter gases diffuse faster.
Worked Example: Graham's Law
How many times faster does H$$_2$$ ($$M = 2$$) diffuse compared to O$$_2$$ ($$M = 32$$)?
$$\dfrac{r_{\text{H}_2}}{r_{\text{O}_2}} = \sqrt{\dfrac{32}{2}} = \sqrt{16} = 4$$
H$$_2$$ diffuses 4 times faster than O$$_2$$.
Kinetic Molecular Theory Formulas
Three Types of Molecular Speed
- Root Mean Square (RMS) Speed: $$$v_{\text{rms}} = \sqrt{\frac{3RT}{M}}$$$
- Average Speed: $$$v_{\text{avg}} = \sqrt{\frac{8RT}{\pi M}}$$$
- Most Probable Speed: $$$v_{\text{mp}} = \sqrt{\frac{2RT}{M}}$$$
where $$R = 8.314$$ J/mol·K, $$M$$ = molar mass (in kg/mol).
Ratio of Molecular Speeds
$$$v_{\text{mp}} : v_{\text{avg}} : v_{\text{rms}} = 1 : 1.128 : 1.224 = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3}$$$
Order: $$v_{\text{mp}} < v_{\text{avg}} < v_{\text{rms}}$$
Worked Example: RMS Speed
Find the RMS speed of N$$_2$$ molecules at $$27°$$C.
$$M_{\text{N}_2} = 28 \; \text{g/mol} = 0.028 \; \text{kg/mol}$$, $$T = 300 \; \text{K}$$
$$v_{\text{rms}} = \sqrt{\dfrac{3 \times 8.314 \times 300}{0.028}} = \sqrt{267{,}236} = \textbf{517 m/s}$$
Average Kinetic Energy
- Per molecule: $$KE = \dfrac{3}{2} k_B T$$
- Per mole: $$KE = \dfrac{3}{2} RT$$
Key insight: Average KE depends only on temperature, not on the type of gas.
Real Gases Formulas
Van der Waals Equation
$$$\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$$$
For 1 mole of gas:
$$$\left(P + \frac{a}{V^2}\right)(V - b) = RT$$$
- $$a$$ = measure of intermolecular attraction (units: atm·L$$^2$$·mol$$^{-2}$$). Higher $$a$$ → stronger attraction → easier to liquefy.
- $$b$$ = measure of molecular volume (units: L·mol$$^{-1}$$). Higher $$b$$ → larger molecules.
Compressibility Factor
$$$Z = \frac{PV}{nRT}$$$
- $$Z = 1$$: gas behaves ideally
- $$Z < 1$$: gas is more compressible than ideal (attractive forces dominate)
- $$Z > 1$$: gas is less compressible than ideal (molecular volume dominates)
Tip: H$$_2$$ and He show $$Z > 1$$ at almost all pressures (very weak intermolecular forces). Other gases like CO$$_2$$, NH$$_3$$ show $$Z < 1$$ at moderate pressures before rising above 1 at high pressures.
Boyle Temperature
$$$T_B = \frac{a}{Rb}$$$
At $$T = T_B$$, a real gas obeys the ideal gas equation over an appreciable range of pressure.
Critical Constants and Liquefaction Formulas
Critical Constants from Van der Waals Constants
$$$T_c = \frac{8a}{27Rb}, \qquad P_c = \frac{a}{27b^2}, \qquad V_c = 3b$$$
Useful relationship: $$\dfrac{P_c V_c}{RT_c} = \dfrac{3}{8} = 0.375$$
Worked Example: Critical Temperature
For CO$$_2$$: $$a = 3.59$$ atm·L$$^2$$/mol$$^2$$, $$b = 0.0427$$ L/mol. Find $$T_c$$.
$$T_c = \dfrac{8 \times 3.59}{27 \times 0.0821 \times 0.0427} = \dfrac{28.72}{0.09463} = 303.5 \; \text{K} \approx 30.4°\text{C}$$
CO$$_2$$ cannot be liquefied above $$30.4°$$C.
Liquid State Properties Formulas
Surface Tension
$$$\gamma = \frac{F}{l} = \frac{\text{Force}}{\text{Length}}$$$
Units: N/m or J/m$$^2$$. Surface tension decreases with increasing temperature.
Viscosity
$$$F = \eta A \frac{dv}{dx}$$$
where $$F$$ = viscous force, $$A$$ = area, $$\dfrac{dv}{dx}$$ = velocity gradient. SI unit: Pa·s. Viscosity of liquids decreases with temperature; viscosity of gases increases with temperature.
Tip: Temperature increase → vapour pressure increases, surface tension decreases, viscosity of liquid decreases. All three trends are caused by increased molecular kinetic energy overcoming intermolecular forces.
Quick Reference: Gas Laws and Kinetic Theory
Summary of Key Formulas
| Law / Quantity | Formula |
|---|---|
| Boyle's Law | $$P_1V_1 = P_2V_2$$ (const. $$T$$, $$n$$) |
| Charles's Law | $$V_1/T_1 = V_2/T_2$$ (const. $$P$$, $$n$$) |
| Gay-Lussac's Law | $$P_1/T_1 = P_2/T_2$$ (const. $$V$$, $$n$$) |
| Ideal Gas Equation | $$PV = nRT$$ |
| Dalton's Law | $$P_{\text{total}} = \sum p_i$$, $$p_i = x_i P_{\text{total}}$$ |
| Graham's Law | $$r_1/r_2 = \sqrt{M_2/M_1}$$ |
| $$v_{\text{rms}}$$ | $$\sqrt{3RT/M}$$ |
| $$v_{\text{avg}}$$ | $$\sqrt{8RT/\pi M}$$ |
| $$v_{\text{mp}}$$ | $$\sqrt{2RT/M}$$ |
| Average KE per mole | $$\frac{3}{2}RT$$ |
| Van der Waals | $$(P + an^2/V^2)(V - nb) = nRT$$ |
| Compressibility Factor | $$Z = PV/nRT$$ |
| Boyle Temperature | $$T_B = a/(Rb)$$ |
States of Matter Formulas for JEE 2027: Conclusion
Revising States of Matter Formulas for JEE 2027 can help students strengthen their understanding of gas laws, the ideal gas equation, kinetic molecular theory and real gas behaviour. Learning the meaning of each quantity, checking units and understanding when an equation applies make it easier to approach numerical problems accurately. The worked examples in this guide connect these formulas with practical calculations.
For effective revision, organise gas laws formulas, molecular speed equations and liquid state properties into a short reference sheet. Practise questions regularly, review mistakes and use the JEE Mains Chemistry Formula PDF 2027 linked above to revisit key relationships. Combining formula revision with problem-solving practice can help build confidence in Chemistry calculations.
