Question

An ideal gas is allowed to expand both reversibly and irreversibly in an isolated system. If ${T_i}$​ is the initial temperature and ${T_f}$​ is the final temperature, which of the following statement is correct?
 A. \[{({T_f})_{rev}} = {({T_f})_{irrev}}\]
B. \[{T_f} = {T_{i\;}}\]for both reversible and irreversible process
C. \[{({T_f})_{irrev}} > {({T_f})_{rev}}\]
D. \[{T_f} > {T_{i\;}}\]for both reversible process but \[{T_f} = {T_i}\] irreversible process

Answer 1

Hint: To answer this question, you should recall the concept of expansion of a gas. Apply the first law of thermodynamics to compare the work done for reversible and irreversible expansion. Now compare the work done obtained to differentiate in temperature in both the cases.

Complete Step by step solution:
We know that according to the first law of thermodynamics we can write a relation for the internal energy of the system as \[\Delta E = \Delta q + \Delta w\] --(i) where
\[\Delta E\]=Change in internal energy of the system
\[\Delta q\]= change in heat energy
\[\Delta w\]= work done
Since it is mentioned in the question that the system is isolated, therefore we can that the change takes place adiabatically. Thus, \[\Delta q\]=0. Using this in the equation we get, \[\Delta E = \Delta w\] --(ii)
We know from the concept of expansion of gas that if gas was to expand by a certain volume reversibly, then it would do a work on the surroundings. If the gas expands irreversibly it would have to do the same amount of work on the surroundings to expand in volume, but it would also have to do work against frictional forces. Hence, the amount of irreversible work is greater in magnitude than reversible work.
\[\therefore \Delta {w_{irreversible}} > \Delta {w_{reversible}}\]
Using this relation in equation (ii) implies,
\[\Delta {E_{irreversible}} > \Delta {E_{reversible}}\]​ ---(iii).
We know that another formula for calculating internal energy is
\[\Delta E = nR\Delta T\]
$\Delta E$= change in internal energy, $n$= no. of moles, $R$= Universal gas constant, $\Delta T$= Change in temperature.
We can see $\Delta E$is directly proportional to $\Delta T$. Comparing this to equation (iii), we can say:
\[{({T_f})_{irrev}} > {({T_f})_{rev}}\]​
Therefore, we can conclude that the correct answer to this question is option C.

Note: The student should not confuse between other types of processes:
Isothermal process: When the system undergoes change from one state to the other, but its temperature remains constant, the system is said to have undergone an isothermal process
Adiabatic process: The process, during which the heat content of the system or a certain quantity of the matter remains constant, is called an adiabatic process.
Isochoric process: The process, during which the volume of the system remains constant, is called an isochoric process. Heating of gas in a closed cylinder is an example of the isochoric process.
Isobaric process: The process during which the pressure of the system remains constant is called an isobaric process.