Adiabatic Process Formula, Derivation & Solved Numericals
An adiabatic process is a key thermodynamic concept where a system changes state without any transfer of heat to or from its surroundings. This does not mean temperature remains unchanged; rather, any change in the system’s internal energy comes only from work done, not from heat.
Adiabatic processes are important in practical applications like compressors, engines, and rapid gas expansions. In compressor theory and related fields, “adiabatic” (no heat transfer) and “isentropic” (constant entropy) are often used interchangeably.
When the process is both adiabatic and reversible, it is precisely called isentropic. This distinction becomes especially relevant in fields like energy engineering, power generation, and renewable energy systems.
Adiabatic Process: Fundamental Explanation
During an adiabatic process, heat transfer (Q) is zero. For an ideal gas, this means any work done by or on the gas translates directly to a change in internal energy (ΔU). Expansion leads to a decrease in temperature, while compression leads to a rise in temperature.
Common examples include:
- Rapid compression in air compressors (output air is warmer than input due to adiabatic heating).
- Sudden opening of pressurized containers (adiabatic cooling causes temperature drops).
- Engine cylinder compressions and gas turbines.
Key Formulas for Adiabatic Processes
| Formula | Description |
|---|---|
| Q = 0 | No heat transfer during the process |
| ΔU = −W | Change in internal energy equals negative of work done (by the gas) |
| PVγ = Constant | Adiabatic relation for pressure and volume (γ = Cp/Cv) |
| T1V1γ−1 = T2V2γ−1 | Temperature-volume relation in adiabatic process |
| W = nCv(T1 − T2) | Work done by the gas (for expansion, W is positive) |
Comparison: Adiabatic vs Isothermal Processes
| Property | Adiabatic | Isothermal |
|---|---|---|
| Heat Transfer (Q) | 0 (no heat exchange) | Not zero |
| Equation | PVγ = constant | PV = constant |
| Internal Energy Change (ΔU) | Non-zero | Zero |
| Work Done | Comes from internal energy change | Balanced by heat flow |
| P-V Graph Shape | Steeper curve | Flatter curve |
Step-by-Step Approach to Adiabatic Problems
To solve typical adiabatic process questions:
- Identify if heat transfer (Q) is zero and confirm initial and final pressure, volume, or temperature data.
- Apply PVγ = Constant to relate pressure and volume, or use the temperature-volume relation.
- If work done is required, use W = nCv(T1 − T2) or an equivalent formula.
- Always use absolute (Kelvin) values for temperature calculations.
Example Problem: Adiabatic Compression in an Engine
Suppose air in a cylinder is compressed adiabatically from an initial pressure of 1.10 × 105 Pa and temperature 300 K, with a compression ratio (V1/V2) of 15:1. The specific heat ratio γ for air (diatomic gas) is 1.4.
To find final pressure (P2) and temperature (T2):
| Step | Formula | Calculation |
|---|---|---|
| Final Pressure | P2 = P1(V1/V2)γ | P2 = 1.10 × 105 × 151.4 (calculate for result) |
| Final Temperature | T2 = T1(V1/V2)γ−1 | T2 = 300 × 150.4 (calculate for result) |
These steps show how compression in diesel engines leads to very high pressures and temperatures, explaining spontaneous fuel ignition.
Practical Considerations in Adiabatic Processes
Adiabatic processes are idealizations, as perfect insulation is not possible. However, fast processes such as gas expansion or compression often approach adiabatic conditions because there isn’t enough time for heat exchange. This is why rapid gas expansion in decompressed spray cans feels cold and compressor outlets feel hot.
A reversible adiabatic process is also called an isentropic process (constant entropy), while real-world adiabatic processes may involve some irreversibility.
Key Application Areas
| Area | Adiabatic Role |
|---|---|
| Compressors (Engines, Turbines) | Predict temperature/pressure changes, calculate work required for compression |
| Industrial Gas Expansion | Explains cooling during rapid expansion in valves or spray nozzles |
| Renewable Energy Systems | Analyzing gas compression/expansion stages in storage or turbines |
Relevant Vedantu Resources and Next Steps
- Thermodynamic Processes – Detailed Guide
- Isothermal vs Adiabatic Processes – Key Differences
- Adiabatic Process – Formula Derivation
- Work Done in Thermodynamics
Mastering adiabatic processes helps in understanding real-world applications in engineering and physics. Practice further with example problems, and use Vedantu’s resources for stepwise derivations and advanced applications.
FAQs on Adiabatic Process in Thermodynamics: Meaning, Formulas & Examples
1. What is an adiabatic process?
An adiabatic process is a thermodynamic process in which no heat is transferred to or from the system (q = 0). The change in the system is due to work done by or on the system, which leads to a change in internal energy and temperature, even though there is no heat exchange with surroundings.
2. What happens during an adiabatic process?
During an adiabatic process:
- No heat enters or leaves the system (q = 0).
- The internal energy change is equal and opposite to the work done by or on the system (ΔU = -W).
- The system’s temperature changes as a direct result of compression (temperature increases) or expansion (temperature decreases).
3. What is the adiabatic process formula?
The standard adiabatic process formula for an ideal gas is:
- PVγ = Constant, where γ = Cp/Cv
- Work done: W = (P1V1 - P2V2)/(γ - 1)
4. Does adiabatic mean q = 0?
Yes, in an adiabatic process, q = 0. This means that there is no heat transfer into or out of the system during the process; all energy change happens by work only.
5. What is the difference between adiabatic and isothermal processes?
Key differences:
- Adiabatic process: No heat exchange (q = 0), temperature changes, PVγ = constant.
- Isothermal process: Temperature remains constant, heat exchange occurs (q ≠ 0), PV = constant.
- In a P-V diagram, the adiabatic curve is steeper than the isothermal curve passing through the same point.
6. Is entropy constant in an adiabatic process?
Entropy remains constant only in a reversible adiabatic process (called isentropic). In an irreversible adiabatic process, entropy increases due to internal friction or other non-ideal factors.
7. How do you calculate work done in an adiabatic process?
To calculate work done (W) in an adiabatic process for an ideal gas:
- Use: W = (P1V1 - P2V2)/(γ - 1)
- Alternatively, if temperatures are known: W = nCv(T1 - T2)
8. What are some real-life examples of adiabatic processes?
Examples of adiabatic processes:
- Rapid compression or expansion of air in a piston or cylinder
- Formation of clouds due to adiabatic cooling as air rises in the atmosphere
- Air pumped out of a tire or spray can (expands adiabatically, feels cold)
- Compression stroke in a diesel engine (temperature rises due to adiabatic compression)
9. What is the relation between temperature and volume in an adiabatic process?
For an adiabatic process involving an ideal gas, the temperature and volume follow:
T1V1γ-1 = T2V2γ-1
This shows that during adiabatic expansion, as volume increases, temperature decreases, and vice versa.
10. Can an adiabatic process occur in practice?
A perfectly adiabatic process is idealized, but in practical terms, a process can be approximately adiabatic if it happens fast enough or the system is well-insulated, so that heat exchange is negligible compared to work done.
11. What is the value of γ (gamma) in the adiabatic equation?
γ (gamma) is the ratio of specific heats (Cp/Cv):
- For monatomic gases: γ ≈ 5/3 ≈ 1.67
- For diatomic gases: γ ≈ 7/5 ≈ 1.4
12. Why is the adiabatic curve steeper than the isothermal curve on a P-V diagram?
On a P-V (pressure-volume) diagram, the adiabatic curve is steeper because, in an adiabatic process, no heat is exchanged, so all energy change is internal. This causes pressure to fall faster with increasing volume compared to an isothermal process, where heat exchange maintains constant temperature.
