Question

The final volume (in L) of one mole of an ideal gas initially at ${27^ \circ }C$ and 8.21atmpressure, if it absorbs 420 cal of heat during a reversible isothermal expansion is?

Answer 1

Hint: First calculate the initial volume using ideal gas equation PV = nRT
Then we can use that value in the formula for reversible isothermal expansion for final volume.

Complete step-by-step solution:
The formula for work done in a reversible isothermal expansion of an ideal gas is,
\[\begin{gathered}
  W = nRT\ln \dfrac{{{V_{\left( {final} \right)}}}}{{{V_{\left( {initial} \right)}}}}
  W = 2.303nRT\log \dfrac{{{V_{\left( {final} \right)}}}}{{{V_{\left( {initial} \right)}}}}
 \\

\end{gathered} \](Or)
So, let us first find out the value of initial volume using the ideal gas equation, PV=nRT.
Given the initial pressure is 8.21atm,
Initial temperature is ${27^ \circ }C$ which upon converting to Kelvin scale is 27 + 273K = 300K
Gas constant R (in cal) is 2 and the number of moles, n is one.
Considering Boyle's law, we have P$ \propto $V which implies the formula for work done in Reversible isothermal expansion in the following manner,
$\begin{gathered}
  W = {\kern 1pt} {\kern 1pt} 2.303nRT\log \dfrac{{{P_{\left( {initial} \right)}}}}{{{P_{\left( {final} \right)}}}}
  420 = {\kern 1pt} 2.303 \times {\kern 1pt} 2 \times {\kern 1pt} 300{\kern 1pt} \log \dfrac{{8.21}}{{{P_{\left( {final} \right)}}}}
  \log {\kern 1pt} 8.21 - log{P_{\left( {final} \right)}} = \dfrac{{420}}{{2.303 \times 600}}
 \\
  \log {P_{\left( {final} \right)}} = 0.9143 - {\kern 1pt} 0.304
  {P_{\left( {final} \right)}} = {\kern 1pt} 4.07atm
\end{gathered} $
Now we have final pressure 4.07atm, temperature being isothermal (constant), temperature remains 300k even after expansion. R is 0.0821. To get the final volume let us substitute the values in the ideal gas equation.
PV=nRT
$\begin{gathered}
  4.07 \times V{\kern 1pt} = {\kern 1pt} 1 \times 0.0821{\kern 1pt} \times 300
  V = \dfrac{{24.63}}{{4.07}}
  {V_{\left( {final} \right)}} = 6L
\end{gathered} $

So, the amount of gas which has expanded isothermally is 6litres.


Additional information: Dalton’s law is one of the ideal gas laws in which relation between pressure and volume is given. Where PV=constant which implies pressure and volume are inversely related.

Note: Remember that the heat absorbed has to be taken in positive (+ve) notation. Because we know that the absorption of heat energy is an endothermic reaction.