Question

Two moles of an ideal monatomic gas occupies a volume \[V\] at \[{27^\circ }{\text{C}}\]. The gas expands adiabatically to a volume \[2V\].Calculate (a) the final temperature of the gas and (b) change in its energy.
(A) \[(a){\text{ }}189{\text{K (b)}} - {\text{2}}{\text{.7}}\,{\text{kJ}}\]
(B) \[(a){\text{ }}195{\text{K (b)2}}{\text{.7}}\,{\text{kJ}}\]
(C) \[(a){\text{ }}189{\text{K (b)2}}{\text{.7}}\,{\text{kJ}}\]
(D) \[(a){\text{ }}195{\text{K (b)}} - {\text{2}}{\text{.7}}\,{\text{kJ}}\]

Answer 1

Hint: Adiabatic process happens in a system in which there is no exchange of heat and mass with surroundings. The change in the internal energy is defined as negative of change in the work done in the adiabatic process. Use the relationship between the temperature and volume for adiabatic processes.

Complete step by step answer:(a)
Write down the expression for the adiabatic process in terms of temperature
\[{T_1}{V_1}^{\gamma - 1} = {T_2}{V_2}^{\gamma - 1}\]
Here \[{T_1}\] and \[{V_1}\] are initial temperature and volume of gas respectively. \[{T_2}\] And \[{V_2}\] are initial temperature and volume of gas. \[\gamma \] is adiabatic constant.
Rearrange
\[{T_2} = \dfrac{{{T_1}{V_1}^{\gamma - 1}}}{{{V_2}^{\gamma - 1}}}\]
Substitute \[V\] for\[{V_1}\], \[2V\] for\[{V_2}\], \[\dfrac{5}{3}\] for \[\gamma \] and \[{27^\circ }{\text{C}}\] for \[{T_1}\]
\[{T_2} = \dfrac{{({{27}^\circ }{\text{C}}){{\left( V \right)}^{\dfrac{5}{3} - 1}}}}{{{{\left( {2V} \right)}^{\dfrac{5}{3} - 1}}}} \\
   \Rightarrow ({27^\circ }{\text{C}})\dfrac{1}{{{{\left( 2 \right)}^{\dfrac{5}{3} - 1}}}} \\
 \Rightarrow \left( {{{27}^\circ }{\text{C}} + 373{\text{ K}}} \right)\dfrac{1}{{{{\left( 2 \right)}^{\dfrac{2}{3}}}}} \\
   \Rightarrow 300{\text{ K}}\dfrac{1}{{{{\left( 2 \right)}^{\dfrac{2}{3}}}}} \\
   \Rightarrow 189{\text{ K}} \\
\]
(b)

Write down the expression for the change in internal energy\[\Delta U\] for adiabatic processes.
\[\Delta U = \dfrac{{nR({T_2} - {T_1})}}{{\gamma - 1}}\]
Here, \[n\] is the number of moles and \[R\] is gas constant.
Substitute \[2\] for\[n\], \[8.31\] for\[R\], \[189{\text{K}}\] for\[{T_2}\], \[300{\text{ K}}\] for \[{T_1}\]
\[\
  \Delta U = \dfrac{{2 \times 8.31(189 - 300{\text{ K}}){\text{ }}}}{{\dfrac{5}{3} - 1}} \\
   \Rightarrow {\text{ }} - 2767{\text{ K}} \\
 \]

Note:We have used ideal gas law here because it was mentioned in the question that the gas is ideal. Otherwise, we would have to use real gas laws. If in the question it was mentioned that the process is sudden but not mentioned adiabatic, then also the process would also be adiabatic. The adiabatic process does not involve any transfer of heat to or inside the system. $\gamma$ is the ratio of specific heat at constant pressure to specific heat at constant volume. It is also given by relation called Mayer’s formula