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Calculate the work done when one mole of a perfect gas is compressed adiabatically. The initial pressure and volume of the gas are \[{10^5}N/{m^2}\] and 6L respectively. The final volume of the gas is 2L. the molar specific heat of the gas at constant volume is \[\dfrac{{3R}}{2}\]

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Last updated date: 13th Jun 2024
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Answer
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Hint: An adiabatic process is a process in which heat is not allowed to leave or enter the system (i.e. no heat exchange with the surroundings). In such a process, both pressures of the system will change with the volume.
Formula used: In this solution we will be using the following formulae;
\[P{V^\gamma } = constant\] where \[P\] stands for pressure and \[V\] for volume, \[\gamma \] is the adiabatic constant.
\[\gamma = \dfrac{{{c_p}}}{{{c_v}}}\] where \[{c_p}\] is the specific heat capacity of a gas at constant pressure, and \[{c_v}\] is the specific heat capacity at constant volume.
\[{c_p} - {c_v} = R\] where \[R\] is the molar gas constant.
\[W = \dfrac{{{P_2}{V_2} - {P_1}{V_1}}}{{1 - \gamma }}\] where \[W\] is the work done by a gas in an adiabatic process, the subscript 2 and 1 signifies the final and initial state of the system.

Complete Step-by-Step solution:
For adiabatic process, we have that
\[P{V^\gamma } = constant\] where \[P\] stands for pressure and \[V\] for volume, \[\gamma \] is the adiabatic constant
Hence, by comparison on one state to another, we may have
\[{P_1}{V_2}^\gamma = {P_2}{V_2}^\gamma \]
But \[\gamma = \dfrac{{{c_p}}}{{{c_v}}}\] where \[{c_p}\] is the specific heat capacity of a gas at constant pressure, and \[{c_v}\] is the specific heat capacity at constant volume.
and again, \[{c_p} - {c_v} = R\] where \[R\] is the molar gas constant.
Hence, by inserting values
\[{c_p} - \left( {\dfrac{{3R}}{2}} \right) = R\]
\[ \Rightarrow {c_p} = R + \dfrac{{3R}}{2} = \dfrac{{5R}}{2}\]
Hence, the adiabatic constant can be calculated as
\[\gamma = \dfrac{{{c_p}}}{{{c_v}}} = \dfrac{{5R}}{2} \div \dfrac{{3R}}{2}\]
\[ \Rightarrow \gamma = \dfrac{5}{3}\]
Hence inserting into \[{P_1}{V_2}^\gamma = {P_2}{V_2}^\gamma \], we have
\[\left( {{{10}^5}} \right){\left( 6 \right)^{\dfrac{5}{3}}} = {P_2}{\left( 2 \right)^{\dfrac{5}{3}}}\]
Hence, by dividing both sides by \[{\left( 2 \right)^{\dfrac{5}{3}}}\] we have
\[{P_2} = \left( {{{10}^5}} \right){\left( 3 \right)^{\dfrac{5}{3}}} = 6.19 \times {10^5}N/{m^2}\]
The work done in an adiabatic process is given by
\[W = \dfrac{{{P_2}{V_2} - {P_1}{V_1}}}{{1 - \gamma }}\]
Hence, inserting all known values, we get
\[W = \dfrac{{6.19 \times {{10}^5}\left( {2 \times {{10}^{ - 3}}} \right) - {{10}^5}\left( {6 \times {{10}^{ - 3}}} \right)}}{{1 - \dfrac{5}{3}}}\] (since 1000 Litre is 1 cubic metre).
Computing the equation, we have
\[W = - 957J\]
Negative signifies work is done on the system.

Note: For clarity, observe that in the relation \[{P_1}{V_2}^\gamma = {P_2}{V_2}^\gamma \] we do not have to convert to SI units. This is because it leads to a ratio of the volumes and is hence units along with any conversion factor will cancel out eventually.