Behaviour of Gases 🌬️

Gases are like crowds of tiny, fast-moving particles. Because each particle is far from its neighbors, they bump into one another only briefly, making their overall behavior surprisingly simple to describe 🙂. :contentReference[oaicite:0]{index=0}

1. The Basic Temperature–Pressure–Volume Link 🔗

  • At low pressure and high temperature, a gas obeys
    \( P\,V = K\,T \)  (12.1), where K depends on how many molecules N are present. :contentReference[oaicite:1]{index=1}
  • Since K = N kB, the universal Boltzmann constant kB appears:
    \( \dfrac{P_1 V_1}{N_1 T_1} = \dfrac{P_2 V_2}{N_2 T_2} = k_B \)  (12.2). :contentReference[oaicite:2]{index=2}
  • If P, V, and T are the same for two gases, they contain the same number of molecules—Avogadro’s big insight! :contentReference[oaicite:3]{index=3}

2. The Ideal-Gas Picture 🧩

  • The handy, all-purpose relation is
    \( P\,V = \mu R T \)  (12.3), with the number of moles \( \mu \) and gas constant \( R = 8.314\;\text{J mol}^{-1}\text{K}^{-1} \). :contentReference[oaicite:4]{index=4}
  • Moles connect to mass and molecules through
    \( \mu = \dfrac{N}{N_A} = \dfrac{M}{M_0} \)  (12.4). :contentReference[oaicite:5]{index=5}
  • Other useful faces of the same coin:
    \( P\,V = k_B N T \quad\text{or}\quad P = k_B n T \) (with number density \( n = N/V \)). :contentReference[oaicite:6]{index=6}
  • Relating pressure to density:
    \( \rho = \dfrac{P\,M_0}{R\,T} \)  (12.5). :contentReference[oaicite:7]{index=7}
  • An ideal gas obeys these equations perfectly. Real gases come closest when pressure is low and temperature is high. :contentReference[oaicite:8]{index=8}

3. Classic Gas Laws in Action ⚖️

Boyle’s Law (Isothermal) 📉

With temperature fixed, \( P\,V = \text{constant} \). Pressure climbs when volume shrinks and vice-versa. :contentReference[oaicite:9]{index=9}

Charles’ Law (Isobaric) 🌡️

At constant pressure, volume rises in direct proportion to absolute temperature: \( V \propto T \). :contentReference[oaicite:10]{index=10}

Dalton’s Law of Partial Pressures 🧮

For a mix of non-reacting ideal gases:

  • \( P\,V = (\mu_1 + \mu_2 + \dots)\,R\,T \)  (12.7) :contentReference[oaicite:11]{index=11}
  • Total pressure is simply \( P = P_1 + P_2 + \dots \)  (12.9). :contentReference[oaicite:12]{index=12}

4. Quick Calculations & Fun Facts ✏️

  • How “empty” is steam? Water vapor at 100 °C and 1 atm has molecular volume only \( 6\times10^{-4} \) of its container—so most of the space is just, well, space! :contentReference[oaicite:13]{index=13}
  • Size of a water molecule: Volume \( \approx 3\times10^{-29}\,\text{m}^3 \), giving a radius of about 2 Å (that’s two-ten-billionths of a meter). :contentReference[oaicite:14]{index=14}
  • Average gap between water molecules in steam: roughly 40 Å—wide enough to fit twenty molecules of water side-by-side! :contentReference[oaicite:15]{index=15}
  • Mixing neon & oxygen: If \( P_{\text{Ne}}:P_{\text{O}_2}=3:2 \), then • number of molecules \( N_{\text{Ne}}:N_{\text{O}_2}=3:2 \); • mass-density ratio \( \rho_{\text{Ne}}:\rho_{\text{O}_2}\approx0.947 \). :contentReference[oaicite:16]{index=16}

5. High-Yield NEET Nuggets 🎯

  1. Ideal-gas equation in its many forms—know how to switch between them quickly.
  2. Boyle’s and Charles’ laws: recognize the inverse and direct relationships on sight.
  3. Dalton’s law of partial pressures—especially useful for multi-gas problems.
  4. Avogadro’s hypothesis & mole concept (22.4 L at STP).
  5. Boltzmann constant \( k_B \) and its bridge between microscopic and macroscopic worlds.

Keep these ideas handy, practice applying the formulas, and you’ll breeze through gas-law questions 🚀!