Kinetic Theory of an Ideal Gas 🧪
1 · Molecular Picture 🚀
- A gas is a huge crowd of molecules (about Avogadro’s number) zooming around in random directions.
- Their average spacing is ≈ 10 times the molecule’s own size, so they barely notice each other except during quick, elastic “bumps” (collisions).
- Every bump with the container’s wall flips the normal component of momentum, creating a steady push we call pressure. :contentReference[oaicite:0]{index=0}
2 · Pressure–Speed Connection 💡
When we add up all the tiny momentum jolts on one wall, we get
\[
P \;=\; \frac{1}{3}\,n\,m\,\overline{v^{2}},
\]
where
\(n\) = molecules per unit volume,
\(m\) = mass of each molecule,
\(\overline{v^{2}}\) = the average of the squared speeds. :contentReference[oaicite:1]{index=1}
3 · Linking Temperature to Motion 🌡️→🏃♂️
Re-packing the ideal-gas equation with the result above gives \[ \frac{1}{2}\,m\,\overline{v^{2}} \;=\; \frac{3}{2}\,k_{B}\,T, \] so the average kinetic energy per molecule depends only on the absolute temperature—nothing else! :contentReference[oaicite:2]{index=2}
4 · Root-Mean-Square Speed 🏎️
Define \[ v_{\text{rms}} \;=\;\sqrt{\overline{v^{2}}}. \] At \(T = 300\;\text{K}\) a nitrogen molecule has \(v_{\text{rms}} \approx 516\;\text{m s}^{-1}\)—about the speed of sound! Lighter molecules dash faster; heavier ones plod along. :contentReference[oaicite:3]{index=3}
5 · Mixtures & Partial Pressures 🥤
For a blend of non-reacting ideal gases, \[ P \;=\; \bigl(n_{1}+n_{2}+\dots\bigr)k_{B}T, \] matching Dalton’s law: each gas adds its own share to the total pressure. All species share the same average kinetic energy because they share the same \(T\). :contentReference[oaicite:4]{index=4}
6 · Law of Equipartition of Energy 🎯
- Every independent “quadratic” motion term (called a degree of freedom) gets \(\tfrac{1}{2}k_{B}T\) of energy on average.
- A free molecule has 3 translational degrees; a rigid diatomic has 3 translational + 2 rotational = 5, and so on.
- Total average energy = \[ \varepsilon_{\text{avg}} = \tfrac{f}{2}\,k_{B}T, \] where \(f\) is the active degree-of-freedom count. :contentReference[oaicite:5]{index=5}
7 · Quick Concept Checks ✅
- Compression warms gas: A piston moving inward acts like a “bat” coming toward molecules; rebounds are snappier, boosting their speed → higher \(T\). 🏏🔥 :contentReference[oaicite:6]{index=6}
- Effusion/enrichment: Lighter \(\mathrm{UF}_6\) with \(^{235}\mathrm{U}\) leaks through pores slightly faster (≈ 0.44 % speed edge) than the heavier \(^{238}\mathrm{U}\) partner. 🔬 :contentReference[oaicite:7]{index=7}
High-Yield NEET Nuggets 🎯
- \(P = \tfrac{1}{3}n m \overline{v^{2}}\) (kinetic-theory pressure formula)
- Average kinetic energy per molecule is \(\tfrac{3}{2}k_{B}T\)
- \(v_{\text{rms}} = \sqrt{\tfrac{3k_{B}T}{m}}\) ➜ lighter gases move faster
- Dalton’s law from kinetic theory: \(P = \bigl(n_{1}+n_{2}+\dots\bigr)k_{B}T\)
- Equipartition: each degree of freedom carries \(\tfrac{1}{2}k_{B}T\) of energy
✨ Keep practicing problems—your grasp will get stronger every day! ✨
