Internal Energy of an Ideal Gas
For a fixed amount of ideal gas with temperature-independent molar heat capacity, internal energy is a state function satisfying ΔU = nC_VΔT; therefore any isothermal ideal-gas process has ΔU = 0.
Why this shows up in the exam
Comparing two paths between the same states · Heating gas in a rigid vessel · Computing temperature change from adiabatic work
Learn the idea
For an ideal gas, internal energy depends only on temperature, not directly on pressure or volume. Ideal-gas molecules store microscopic kinetic energy, so equal temperature changes give equal internal-energy changes even along different paths.
🧠 Memory hook: Ideal-gas U follows T, whatever path P and V take.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- ΔU = n C_V (T_f - T_i) — ideal-gas internal-energy change when C_V is constant over the interval
- dU = n C_V dT — differential form for a calorically perfect ideal gas
How to approach it
- 1Find endpoint temperatures
- 2Use C_V for internal energy
- 3Check that a complete cycle has net ΔU equal to zero
Common slip-ups that cost marks
- •Setting ΔU equal to zero for every cycle leg
- •Using C_P instead of C_V in the state-function relation
- •Inferring ΔU from work alone without the first law
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A gas absorbs 500 J of heat and does 200 J of work. What is the change in its internal energy?
More from Thermodynamics
Thermodynamic Processes and P-V Diagrams
Study different thermodynamic processes (isothermal, isobaric, isochoric, adiabatic, polytropic, cyclic), their definitions, characteristics, and graphical representation on P-V diagrams.
Laws of Thermodynamics
Understand the zeroth and first laws of thermodynamics, including their statements, implications, and applications to physical systems.
Internal Energy, Heat, and Work
Explore the concepts of internal energy, heat, and work, including their definitions, relationships, and how they change during various thermodynamic processes.
Ideal Gas Law and Equation
Learn the ideal gas equation, its relation to physical quantities like pressure, volume, temperature, and density, and its use in describing the behavior of ideal gases.
Gibbs Free Energy and Spontaneity
Learn how Gibbs free energy determines spontaneity, how to calculate it, and its dependence on temperature, pressure, and other thermodynamic parameters.
Entropy and Its Changes
Understand entropy as a measure of disorder, how it changes in physical and chemical processes, and its calculation in various scenarios including phase transitions and isothermal processes.