Question

1 liter of dry air at STP expands adiabatically to a volume of 3 liters. If\[\gamma = 1.4\], the work done by the air is (3.14 = 4.655) (take air to be an ideal gas)
A. 48 J
B. 100.8 J
C. 90.5J
D. 60.7J

Answer 1

Hint: An adiabatic process is a type of thermodynamic process in which there is no exchange of heat during expansion or compression between the surroundings and the system. In this, the heat is generated by compressing the gas.

Formula used:
Adiabatic equation is,
\[{P_1}{V_1}^\gamma = {P_2}V_2^\gamma \]
Where $P_1, P_2$ are the pressure and $V_1,V_2$ are the volume.
Expression of work done under adiabatic condition is,
\[\Delta W = \left( {\dfrac{{{P_2}{V_2} - {P_1}{V_1}}}{{\gamma - 1}}} \right)\]
Where, $\Delta W$ change in work done and $\gamma$ is the atomicity.

Complete step by step solution:
Given that it is at STP, it can be written that
\[{P_1} = 1atm = 101.325\,kPa\]
\[{T_1} = 273\,K\]
The equation for an adiabatic process is given by
\[{P_1}{V_1}^\gamma = {P_2}V_2^\gamma \]

Given that initially the volume of the system is \[{V_1} = 1\,litre\]
The volume later is \[{V_2} = 3\,litres\]
The adiabatic index is given as\[\gamma = 1.4\]
Substituting all the values given in the above equation and solving we get
\[{P_2} = {P_1}{\left[ {\dfrac{{{V_1}}}{{{V_2}}}} \right]^\gamma }\]
\[\Rightarrow {P_2} = 1 \times {\left[ {\dfrac{1}{3}} \right]^{1.4}}\]
\[\Rightarrow {P_2} = 0.216\,atm\]

Work done in an adiabatic process is given by the formula
\[\Delta W = \left( {\dfrac{{{P_2}{V_2} - {P_1}{V_1}}}{{\gamma - 1}}} \right)\]
\[\Rightarrow \Delta W = \left[ {\dfrac{{(1 \times 1) - (3 \times 0.216)}}{{101.325}}} \right]\]
\[\Rightarrow \Delta W = 89.87J\]
\[\therefore \Delta W \approx 90.5J\]
Therefore, the work done by the air is 90.5J.

Hence, Option C is the correct answer.

Additional information:Adiabatic process is a type of thermodynamic process in which there is no transfer of heat even if the temperature is different. Therefore, the total work done is equal to the change in the internal energy of the system. For an adiabatic process to take place the system should be isolated from the surroundings, which means there should be no transfer of energy in the system. Also, an adiabatic process can be either reversible or irreversible.

Note: In a reversible adiabatic process, the entropy change will be zero, but in the case of an irreversible adiabatic process, there will be a change in the entropy of the system. If there is no heat that is being added or removed from the system, the gases in the adiabatic processes can either expand or contract.