Ideal-Gas Equation & Absolute Temperature 🤓
Let’s explore how gases behave at low density and how this leads to an absolute temperature scale that starts at 0 K. Buckle up—this is surprisingly fun! 🚀
1. How a Low-Density Gas Behaves
- The key variables are pressure (P), volume (V) and absolute temperature (T), where T = t + 273.15 K; here t is temperature in °C. :contentReference[oaicite:0]{index=0}
- Boyle’s law: keeping T fixed gives \(P\,V = \text{constant}\). 🌡️ :contentReference[oaicite:1]{index=1}
- Charles’ law: keeping P fixed gives \(\dfrac{V}{T} = \text{constant}\). 📈 :contentReference[oaicite:2]{index=2}
- Combining them shows that \(\dfrac{P\,V}{T} = \text{constant}\) for any low-density gas. :contentReference[oaicite:3]{index=3}
2. The Ideal-Gas Equation
The combined relationship becomes the famous ideal-gas equation:
\(P\,V = \mu R T\) (10.2)
- \(\mu\) = number of moles of gas
- \(R = 8.31\;\text{J mol}^{-1}\text{ K}^{-1}\) (universal gas constant) 🔧
If the volume stays fixed, then \(P \propto T\). That’s the working principle of a constant-volume gas thermometer. :contentReference[oaicite:4]{index=4}
3. Constant-Volume Gas Thermometer
- With V held constant, plotting pressure against temperature gives a straight line. A real-world thermometer simply measures the pressure of a trapped gas to read T. :contentReference[oaicite:5]{index=5}
- Extrapolating that straight line suggests pressure would hit zero at a very chilly –273.15 °C. ❄️
4. Absolute Zero & Kelvin Scale ❄️
- The temperature –273.15 °C is called absolute zero (0 K). No gas can have a lower temperature. :contentReference[oaicite:6]{index=6}
- The Kelvin scale starts here and uses the same-sized degree as the Celsius scale.
- Relationship between the two scales: \(T = t_C + 273.15\) (10.3) 🔥 :contentReference[oaicite:7]{index=7}
5. Switching Between Fahrenheit & Celsius 🌡️
To hop between the two older scales, use
\(\displaystyle \frac{t_F – 32}{180} = \frac{t_C}{100}\) (10.1)
Quick check: 0 °C plugs in to give 32 °F—exactly what you’d expect. :contentReference[oaicite:8]{index=8}
Quick Checks 💡
- If you double the absolute temperature (T) of a gas at constant pressure, its volume doubles. 🎈
- Why does a gas thermometer agree no matter which gas you use? All low-density gases expand the same way, so their P–T plots overlap. 👍
High-Yield Ideas for NEET 🔥
- Boyle’s law: \(P\,V = \text{constant}\) for isothermal changes. :contentReference[oaicite:9]{index=9}
- Charles’ law: \(V/T = \text{constant}\) for isobaric changes. :contentReference[oaicite:10]{index=10}
- Ideal-gas equation \(P\,V = \mu R T\) plus the value of \(R\). :contentReference[oaicite:11]{index=11}
- Absolute zero (–273.15 °C) and the Celsius-to-Kelvin link \(T = t_C + 273.15\). :contentReference[oaicite:12]{index=12}
- Celsius–Fahrenheit conversion trick for quick unit changes. :contentReference[oaicite:13]{index=13}
You’ve got this! Keep practicing, and these gas laws will feel as natural as breathing. 🌬️😃
