1 for an isobaric process, \(\mathrm{\mathrm{q}_{\mathrm{p}}=\Delta \mathrm{U}+\mathrm{w}}\)
2 for an adiabatic process, \(\mathrm{\Delta \mathrm{U}=-\mathrm{W}}\)
3 for an isochoric process, \(\mathrm{\Delta \mathrm{U}=-\mathrm{q}_{\mathrm{v}}}\)
4 for an isothermal process, \(\mathrm{\mathrm{q}=+\mathrm{w}}\)
Explanation:
In isobaric process, pressure is constant and volume increases from \(\mathrm{\mathrm{V}_{1}}\) to \(\mathrm{\mathrm{V}_{2}}\) due to the absorption of heat. According to the first law of thermodynamics, \(\mathrm{\Delta \mathrm{U}=\mathrm{q}+\mathrm{W}}\)
At constant pressure, \(\mathrm{\mathrm{q}_{\mathrm{p}}=\Delta \mathrm{U}-\mathrm{W}}\)
\({{\rm{q}}_{\rm{p}}}{\rm{ = }}\Delta {\rm{U - p}}\Delta {\rm{V}}\)
\({{\rm{q}}_{\rm{p}}}{\rm{ = }}{{\rm{U}}_{\rm{2}}}{\rm{ - }}{{\rm{U}}_{\rm{1}}}{\rm{ - }}\left( {{\rm{ - p}}\left( {{{\rm{V}}_{\rm{2}}}{\rm{ - }}{{\rm{V}}_{\rm{1}}}} \right)} \right)\)
\({{\rm{q}}_{\rm{p}}}{\rm{ = }}\left( {{{\rm{U}}_{\rm{2}}}{\rm{ + p}}{{\rm{V}}_{\rm{2}}}} \right){\rm{ - }}\left( {{{\rm{U}}_{\rm{1}}}{\rm{ + p}}{{\rm{V}}_{\rm{1}}}} \right)\)
\({{\rm{q}}_{\rm{p}}}{\rm{ = }}{{\rm{H}}_{\rm{2}}}{\rm{ - }}{{\rm{H}}_{\rm{1}}}\)
\({{\rm{q}}_{\rm{p}}}{\rm{ = }}\Delta {\rm{H}}\,\,\,\,\,\,\,\,\,\,{\rm{(1)}}\)
As we know, \(\mathrm{\Delta \mathrm{H}=\Delta \mathrm{U}+\mathrm{p} \Delta \mathrm{V}}\)
\(\mathrm{\Delta \mathrm{H}=\Delta \mathrm{U}+\mathrm{W}}\)
Putting the value of \(\Delta {\rm{H}}\,\,in\,\,(1)\)
\(\rm{\mathrm{q}_{\mathrm{p}}=\Delta \mathrm{U}+\mathrm{W}}\)
Other corrected options are as follows:
\( \bullet \) For adibatic process, \(\mathrm{\Delta \mathrm{U}=\mathrm{W}}\)
\( \bullet \) For an isochoric process, \(\mathrm{\Delta \mathrm{U}=\mathrm{q}_{\mathrm{v}}}\)
\( \bullet \) For an isothermal process \(\mathrm{\mathrm{q}=-\mathrm{W}}\)