Explain in detail an adiabatic process.

Adiabatic process: This is a process in which no heat flows into or out of the system $(\mathrm{Q}=0)$. But the gas can expand by spending its internal energy or gas can be compressed through some external work. So the pressure, volume and temperature of the system may change in an adiabatic process. For an adiabatic process, the first law becomes $\Delta \mathrm{U}=\mathrm{W}$.
This implies that the work is done by the gas at the expense of internal energy or work is done on the system which increases its internal energy.

The adiabatic process can be achieved by the following methods:
(i) Thermally insulating the system from surroundings so that no heat flows into or out of the system; for example, when thermally insulated cylinder of gas is compressed (adiabatic compression) or expanded (adiabatic expansion) as shown in the Figure.
(ii) If the process occurs so quickly that there is no time to exchange heat with surroundings even though there is no thermal insulation. A few examples are shown in Figure.
The equation of state for an adiabatic process is given by $\mathrm{PV}^\gamma=$ constant
Here $\gamma$ is called adiabatic exponent $\left(\gamma=\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)$ which depends on the nature of the gas. The equation (1) implies that if the gas goes from an equilibrium state $\left(P_i, V_i\right)$ to another equilibrium state $\left(\mathrm{P}_{\mathrm{f}}, V_{\mathrm{f}}\right)$ adiabatically then it satisfies the relation
$
\mathrm{P}_i \mathrm{~V}_i^\gamma=\mathrm{P}_f \mathrm{~V}_f^\gamma
$
The PV diagram for an adiabatic process is also called adiabat. But actually the adiabatic curve is steeper than isothermal curve.
We can also rewrite the equation (1) in terms of $T$ and $V$. From ideal gas equation, the pressure $P=$ $\frac{\mu \mathrm{RT}}{\mathrm{V}}$. Substituting this equation in the equation (1), we have
$
\frac{\mu R T}{\mathrm{~V}} \mathrm{~V}^\gamma=\text { constant (or) } \frac{\mathrm{T}}{\mathrm{V}} \mathrm{V}^\gamma=\frac{\text { constant }}{\mu \mathrm{R}}
$
Note here that is another constant. So it can be written as
$
\mathrm{TV}^{\gamma-1}=\text { constant. }
$
The equation implies that if the gas goes from an initial equilibrium state $\left(T_i, V_i\right)$ to final equilibrium state $\left(T_f, V_f\right)$ adiabatically then it satisfies the relation
$
\mathrm{T}_i \mathrm{~V}_i^{\gamma-1}=\mathrm{T}_f \mathrm{~V}_f^{\gamma-1}
$
The equation of state for adiabatic process can also be written in terms of $\mathrm{T}$ and $\mathrm{P}$ as

$\mathrm{T}^\gamma \mathrm{P}^{1-\gamma}=$ constant.