KAHIBARO
Table of Contents
What makes a process adiabatic
An adiabatic process is a thermodynamic process in which no heat is transferred between the system and its surroundings. In symbols, this means
$$Q = 0$$
This does not mean that the temperature must stay constant. It only means that energy does not enter or leave as heat. The system can still change its internal energy because work can be done on it or by it.
For an adiabatic process, the first law of thermodynamics becomes
$$\Delta U = Q - W = -W$$
if we use the convention that $W$ is the work done by the system. This tells us that when a gas expands adiabatically and does work on its surroundings, its internal energy decreases, so its temperature usually falls. When a gas is compressed adiabatically, work is done on the gas, its internal energy increases, and its temperature usually rises.
For an adiabatic process,
$$Q = 0$$
and therefore
$$\Delta U = -W$$
if $W$ is the work done by the system.
Physical picture
An adiabatic process happens when heat does not have time to flow, or when the system is very well insulated. A rapid compression of air in a bicycle pump is a common example. The air gets hotter because work is done on it, but there is little time for heat to escape. Another example is the expansion of rising air in the atmosphere. As the air expands in lower pressure, it cools.
It is important not to confuse adiabatic with isothermal. In an isothermal process, temperature stays constant. In an adiabatic process, temperature usually changes.
Adiabatic processes for an ideal gas
For an ideal gas, adiabatic changes obey special relations. These are most often used for reversible adiabatic processes, sometimes called quasi-static adiabatic processes.
If $\gamma$ is the heat capacity ratio,
$$\gamma = \frac{C_P}{C_V}$$
then the main adiabatic relation is
$$PV^\gamma = \text{constant}$$
Other equivalent forms are
$$TV^{\gamma - 1} = \text{constant}$$
and
$$T^\gamma P^{1-\gamma} = \text{constant}$$
These formulas connect the pressure, volume, and temperature of an ideal gas as it changes adiabatically.
For a reversible adiabatic process of an ideal gas,
$$PV^\gamma = \text{constant}$$
$$TV^{\gamma - 1} = \text{constant}$$
$$T^\gamma P^{1-\gamma} = \text{constant}$$
These relations are not general for every substance, they are for ideal gases.
How temperature changes
Because no heat enters during expansion, the gas must use its own internal energy to do work. That is why temperature decreases during adiabatic expansion. During adiabatic compression, the opposite happens, and temperature increases.
For an ideal gas, internal energy depends only on temperature, so the connection is direct. If the temperature rises, the internal energy rises. If the temperature falls, the internal energy falls.
A useful form for internal energy change is
$$\Delta U = nC_V \Delta T$$
So for an adiabatic ideal gas process,
$$nC_V \Delta T = -W$$
This gives a simple interpretation. Positive work done by the gas lowers its temperature, and negative work done by the gas raises its temperature.
Work done in an adiabatic process
For a reversible adiabatic process, the work can be found from the pressure volume relation. The result is
$$W = \frac{P_iV_i - P_fV_f}{\gamma - 1}$$
where the subscripts $i$ and $f$ mean initial and final states.
Using the ideal gas law, this can also be written as
$$W = \frac{nR(T_i - T_f)}{\gamma - 1}$$
These formulas show clearly that if the final temperature is lower than the initial temperature, the gas has done positive work.
For a reversible adiabatic process of an ideal gas,
$$W = \frac{P_iV_i - P_fV_f}{\gamma - 1}$$
and
$$\Delta U = -W$$
Adiabatic curve on a pressure-volume diagram
On a $P$ versus $V$ graph, an adiabatic curve is steeper than an isothermal curve through the same point. This is because pressure drops faster with volume during adiabatic expansion, since the gas is cooling at the same time.
In this sketch, both curves pass through the same starting point. The adiabatic curve falls more sharply.
Reversible and irreversible adiabatic processes
An adiabatic process only requires that $Q=0$. It does not automatically mean the process is reversible. A rapid compression, an explosion, or free expansion into a vacuum may be adiabatic but not reversible.
The familiar equations such as $PV^\gamma = \text{constant}$ apply to reversible adiabatic changes of an ideal gas. If the process is irreversible, the simple path relation generally does not apply, although the condition $Q=0$ still does.
This distinction is very important in thermodynamics.
Adiabatic means no heat transfer.
It does not necessarily mean reversible.
The relation $PV^\gamma = \text{constant}$ is for reversible adiabatic ideal gas processes.
Comparison with other common processes
The following table helps separate adiabatic processes from some other standard thermodynamic processes.
| Process type | Heat transfer $Q$ | Temperature | Typical condition |
|---|---|---|---|
| Adiabatic | $0$ | usually changes | insulated or very fast |
| Isothermal | not necessarily $0$ | constant | good thermal contact |
| Isochoric | can be nonzero | may change | constant volume |
| Isobaric | can be nonzero | may change | constant pressure |
This comparison shows why the word adiabatic should be used carefully. It describes heat transfer, not directly temperature, pressure, or volume alone.
Everyday examples
A bicycle pump becomes warm during rapid compression because the air inside undergoes an approximately adiabatic compression. A spray can or gas cylinder can become cold near the valve during rapid release because the gas expands approximately adiabatically. In the atmosphere, rising air expands and cools, while sinking air compresses and warms, often approximately adiabatically over short times.
These examples are only approximate because perfect insulation is difficult. Still, the adiabatic idea is very useful because it captures the main energy change.
Key ideas to remember
An adiabatic process is defined by zero heat transfer. In such a process, changes in internal energy come from work alone. For an ideal gas, adiabatic expansion cools the gas and adiabatic compression heats it. For reversible adiabatic ideal gas processes, the variables satisfy relations such as $PV^\gamma = \text{constant}$.
Main facts for adiabatic processes:
$$Q = 0$$
$$\Delta U = -W$$
For a reversible adiabatic ideal gas,
$$PV^\gamma = \text{constant}$$
Expansion lowers temperature, compression raises temperature.