Question

In an adiabatic expansion of a gas, its temperature :
$\left( {\text{A}} \right)$ Always increase
$\left( {\text{B}} \right)$ Always diminishes
$\left( {\text{C}} \right)$ Remains constant
$\left( {\text{D}} \right)$ None of these

Answer 1

Hint:Here we use the equation of the first law of Thermodynamics.
For the adiabatic process of gas, the heat is kept zero.
In the adiabatic expansion of the gas, the work done by the gas is always positive.
Since the work done by the gas is positive the internal energy decreases, so the temperature of the gas also decreases.

Formula used:
The first law of Thermodynamics, $Q = ({U_f} - {U_i}) + W$,
where $Q = $ Heat gain or loss in a system,
$({U_f} - {U_i}) = $ Change in internal energy of the gas.
$W = $ Work done by the gas.

Complete step by step answer:
The Adiabatic Process is the process where during the change of volume and pressure of gas no heat is taken from surroundings or given to the surrounding. That means the heat gain or loss of a system in this process is kept zero.
The Adiabatic process is of two types - (i) Adiabatic Expansion and (ii) Adiabatic compression.
In the Adiabatic Expansion of the gas, the work done by the gas is always positive i.e. \[W > 0\].
The first law of Thermodynamics states, $Q = ({U_f} - {U_i}) + W$.
where $Q = $ Heat gain or loss in a system,
$({U_f} - {U_i}) = $ Change in internal energy of the gas.
$W = $ Work done by the gas.
Therefore from the First Law of Thermodynamics, we get that the change of internal energy $({U_f} - {U_i}) < 0$ and hence, ${U_f} < {U_i}$.
Now the First Law of Thermodynamics is also represented as $dQ = dU + dW$ and for the constant-volume process \[dW = 0\],
Therefore, $dQ = dU$. If the specific heat at constant volume is \[{C_v}\], then it can be written by its definition
\[{C_v} = \dfrac{{dQ}}{{dT}}\]
\[or,dQ = {C_v}dT\]
\[or,dU = {C_v}dT\]
From this relation, we get that if the \[dU\] is negative i.e $({U_f} - {U_i}) < 0$ and hence, ${U_f} < {U_i}$, the temperature change \[dT\] will also be negative.
So we can say for an adiabatic expansion of gas since the internal energy decreases, the temperature of the gas is also decreased.
Hence the right option is in option $\left( {\text{B}} \right)$.

Notes:The Adiabatic Expansion has another name which is the Adiabatic Cooling process. This is because the temperature decreases.
We can explain this with an example:
Suppose some gas is kept in a pipe of an insulating wall with an active piston. The piston can move easily without friction. Now if we make the gas expand very quickly, the gas will do some work. The required energy for this work done is taken from the internal energy of the gas i.e the internal energy is decreased which results in the temperature decreases. Since the pipe is of an insulating wall and the gas expands very quickly no heat can be entered from surroundings and therefore the gas remains cool.