First Law of Thermodynamics & Heat Capacity 🎈
1. Changing a System’s Internal Energy 🏋️♂️🔥
A gas in a cylinder can gain or lose internal energy in two simple ways :contentReference[oaicite:0]{index=0}:
- Heat flow: Touch the cylinder with something hotter ➜ energy (heat) rushes in.
- Mechanical work: Push the piston down (or let it push up) ➜ work is done on (or by) the gas.
Heat and work are actions, not possessions. They change the gas’s internal energy U, but the gas never “contains” heat or work :contentReference[oaicite:1]{index=1}.
2. Heat ≠ Internal Energy ⚡️
Heat is energy in transit; internal energy U is a property the system always has. That’s why we track U, not “amount of heat.” :contentReference[oaicite:2]{index=2}
3. The First Law of Thermodynamics 🌟
When energy moves by heat or work, it obeys:
\( \Delta Q \;=\; \Delta U \;+\; \Delta W \) (11.1) :contentReference[oaicite:3]{index=3}
Here:
- \(\Delta Q\) = heat supplied to the system
- \(\Delta W\) = work done by the system
- \(\Delta U\) = change in internal energy
\( \Delta Q – \Delta W = \Delta U \) (11.2) :contentReference[oaicite:4]{index=4}
Because U is a state property, \(\Delta U\) depends only on start and finish, never on the path. The combo \(\Delta Q – \Delta W\) is therefore path-independent too :contentReference[oaicite:5]{index=5}.
Cool shortcut: In an isothermal ideal-gas expansion (\(\Delta U = 0\)), heat equals work: \( \Delta Q = \Delta W \) :contentReference[oaicite:6]{index=6}.
4. Work at Constant Pressure 🚀
For a piston pushing against fixed pressure P:
\( \Delta W = P\,\Delta V \) :contentReference[oaicite:7]{index=7}
Plugging this into the First Law gives:
\( \Delta Q = \Delta U + P\,\Delta V \) (11.3) :contentReference[oaicite:8]{index=8}
Example: Boiling 1 g of Water 💧➡️💨
At 1 atm:
- Latent heat: \(\Delta Q = 2256 \text{ J}\)
- Volume jump: 1 cm³ ➜ 1671 cm³
- Work: \( \Delta W = 1.013\times10^{5}\,(1671-1)\times10^{-6} = 169.2 \text{ J}\)
- Internal-energy rise: \( \Delta U = 2256 – 169.2 = 2086.8 \text{ J}\)
5. Heat Capacity 🔥
Heat needed for a temperature boost \(\Delta T\) defines the heat capacity S:
\( S = \dfrac{\Delta Q}{\Delta T} \) (11.4) :contentReference[oaicite:10]{index=10}
For a mass m we like the specific heat capacity \(s\):
\( s = \dfrac{1}{m}\,\dfrac{\Delta Q}{\Delta T} \) (11.5) :contentReference[oaicite:11]{index=11}
Units: J kg⁻¹ K⁻¹.
Using moles (\(\mu\)) gives the molar heat capacity \(C\):
\( C = \dfrac{1}{\mu}\,\dfrac{\Delta Q}{\Delta T} \) (11.6) :contentReference[oaicite:12]{index=12}
Quick check for solids 📦
Each atom vibrates → per mole: \(U = 3RT\). Volume hardly changes, so
\( C \approx 3R \) (11.7) :contentReference[oaicite:13]{index=13}
That’s about 25 J mol⁻¹ K⁻¹—pretty close to many real measurements.
📝 High-Yield Ideas for NEET
- First-Law equation \( \Delta Q = \Delta U + \Delta W \) and sign conventions.
- Difference between heat, work, and internal energy—only U is a state function.
- Work formula \( \Delta W = P\Delta V \) for constant-pressure processes.
- Specific vs. molar heat capacity definitions and units.
- For solids at room temperature, \(C \approx 3R\) (Dulong-Petit law).
🎯 Key Takeaways
- You change a system’s internal energy by heating or doing work.
- The First Law keeps the energy balance honest—nothing appears or disappears.
- Track heat capacity to know how much energy a temperature rise needs.
- Always keep an eye on sign conventions: heat added is positive, work done by the system is positive.
Keep practicing—thermodynamics gets easier with every problem you solve! 🚀✨
