Question 1: Thermodynamics & KTG

Question

Consider one mole of helium gas enclosed in a container at initial pressure P
1
and volume V
1
. It
expands isothermally to volume 4V
1
. After this, the gas expands adiabatically and its volume becomes
32V
1
. The work done by the gas during isothermal and adiabatic expansion processes are W
iso
and
W
adia
, respectively. If the ratio ${{{W_{iso}}} \over {{W_{adia}}}}$ = f ln 2, then f is ______.

Accepted Answer

${{{p_1}} \over 4}{(4{V_1})^{5/3}} = {p_2}{(32{V_1})^{5/3}}$
${p_2} = {{{p_1}} \over 4}{\left( {{1 \over 8}} ight)^{5/3}} = {{{p_1}} \over {128}}$
${W_{adi}} = {{{p_1}{V_1} - {p_2}{V_2}} \over {\gamma  - 1}}$
$ = {{{p_1}{V_1} - {{{p_1}} \over {128}}(32{V_1})} \over {{5 \over 3} - 1}}$
$ = {{{p_1}{V_1}(3/4)} \over {2/3}} = {9 \over 8}{p_1}{V_1}$
${W_{iso}} = {p_1}{V_1}\ln \left( {{{4{V_1}} \over {{V_1}}}} ight) = 2{p_1}{V_1}\ln 2$
$ 	herefore $ ${{{W_{iso}}} \over {{W_{adi}}}} = {{16} \over 9}\ln 2$
$ \Rightarrow f = {{16} \over 9} = 1.7778 \approx 1.78$
![](https://imagex.cdn.examgoal.net/1ks6icwz6/72ef7892-f58d-462f-a793-e085fad4e5e4/6254dd20-fa18-11eb-993d-7f6f2f90a717/file-1ks6icwz7.jpeg)

Thermodynamics & KTG

Integer / Numeric Answer Type