Adiabatic Process

Adiabatic Process: The Foundation

An adiabatic process is one in which no heat flows between the system and its surroundings. The word comes from Greek — adiabatos meaning "impassable" — and it describes a wall that heat cannot cross.

This is a direct consequence of the First Law: $\Delta U = Q + W$. With $Q = 0$, we get $\Delta U = W$ (where $W$ is work done on the system). Many textbooks write work done by the system as positive, giving $\Delta U = -W_{\text{by}}$. Both conventions appear — what matters is the physical idea: no heat exchange means energy change equals work.

A perfectly adiabatic process is an idealisation. In practice, a process is approximately adiabatic if it happens very quickly (so heat has no time to flow) or if the system is very well insulated. The rapid compression of air in a diesel engine's cylinder is a classic example — the temperature rises so sharply that the fuel ignites without a spark plug.

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The Relation Between $P$ and $V$ in an Adiabatic Process

For an ideal gas undergoing a reversible adiabatic process, pressure and volume obey a simple power law. The derivation uses the First Law together with the ideal gas equation and the expression for molar specific heat.

Consider one mole of an ideal gas. In an infinitesimal adiabatic step, $dQ = 0$, so the First Law gives:

For an ideal gas, the internal energy depends only on temperature: $dU = C_V dT$, where $C_V$ is the molar specific heat at constant volume. The work done by the gas in a small volume change is $dW = P dV$. Therefore:

The minus sign appears because we are writing $dW$ as work done by the gas (positive when the gas expands). The First Law in the form $dU = dQ - dW$ (with $dQ = 0$) gives $C_V dT = -P dV$.

Now use the ideal gas equation for one mole: $PV = RT$. Differentiate:

From the first equation, $dT = -\frac{P dV}{C_V}$. Substitute into the differentiated ideal gas equation:

Rearrange:

Bring the $P dV$ terms together:

Recall that $C_P - C_V = R$ for an ideal gas, so $1 + \frac{R}{C_V} = \frac{C_V + R}{C_V} = \frac{C_P}{C_V} = \gamma$, the ratio of specific heats. Thus:

Separate variables:

Integrate both sides:

Exponentiate:

This is the central result. The constant depends on the initial state — if the gas starts at $(P_1, V_1)$ and ends at $(P_2, V_2)$, then:

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Properties Derived from $PV^{\gamma} = \text{constant}$

The book lists three key properties that follow directly from this relation. Each one gives a different way to describe the same adiabatic curve.

Property (I): $TV^{\gamma-1} = \text{constant}$

Start from $PV^{\gamma} = \text{constant}$. Use the ideal gas law $P = \frac{nRT}{V}$ (for $n$ moles). Substitute:

The $nR$ is constant, so:

This tells you how temperature changes when volume changes adiabatically. If the gas expands ($V$ increases), $T$ must drop — the gas cools. If compressed, it heats up.

Property (II): $T^{\gamma} P^{1-\gamma} = \text{constant}$

Again start from $PV^{\gamma} = \text{constant}$. This time eliminate $V$ using $V = \frac{nRT}{P}$:

This form is useful when you know pressure and temperature but not volume.

Property (III): Work Done in an Adiabatic Process

For a finite change from volume $V_1$ to $V_2$, the work done by the gas is:

Since $P V^{\gamma} = K$ (a constant), we have $P = K V^{-\gamma}$. Therefore:

For $\gamma \neq 1$ (which is always true for gases — $\gamma > 1$):

Since $K = P_1 V_1^{\gamma} = P_2 V_2^{\gamma}$, we can write this in several equivalent forms:

The second form is often more convenient. Notice that if the gas expands ($V_2 > V_1$), it cools ($T_2 < T_1$), so $W$ is positive — the gas does work. If compressed, $W$ is negative — work is done on the gas.

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The $P$-$V$ Diagram: Adiabatic vs Isothermal …