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Consider that an ideal gas (n moles) is expanding in a process given by P=f(V), which passes through a point(V_(0),P_(0)). Show that the gas is absorbing heat at (P_(0),V_(0))if the slope of the curve P=f (V) is larger than the slope of the adiabat passing through (P_(0), V_(0)).

Answer» <html><body><p></p>Solution :Slope of `P=f(V)`, curve at `(V_(0) , P_(0))= f(V_(0))` <br/> Slope of <a href="https://interviewquestions.tuteehub.com/tag/adiabat-2400725" style="font-weight:bold;" target="_blank" title="Click to know more about ADIABAT">ADIABAT</a> at `(V_(0) P_(0))` <br/> `=k(-gamma) V_(0)^(-<a href="https://interviewquestions.tuteehub.com/tag/1-256655" style="font-weight:bold;" target="_blank" title="Click to know more about 1">1</a>-gamma)= -gammaP_(0)//V_(0)` <br/> Now heat absorbed in the process P = f(V) <br/> `dQ = dU +dW = nC_(v)dT +PdV` <br/> Since `T=(1//nR) PV=(1//nR) V f(V)` <br/> `dT = (1//nR) [f(V)+Vf.(V)]dV` <br/> Thus `(dQ)/(dV)underset(V=V_(0))("" ) =(CV)/(R )[f(V_(0))+V_(0)f.(V_(0))]+f(V_(0))`<br/> `=[(1)/(gamma-1)+1]f(V_(0)) +(V_(0)f.(V_(0)))/(gamma-1)` <br/> `=(gamma)/(gamma-1)P_(0) +(V_(0))/(gamma-1)f.(V_(0))` <br/> Heat is absorbed when `dQ//dV <a href="https://interviewquestions.tuteehub.com/tag/gt-1013864" style="font-weight:bold;" target="_blank" title="Click to know more about GT">GT</a> 0` when gas <a href="https://interviewquestions.tuteehub.com/tag/expands-2076032" style="font-weight:bold;" target="_blank" title="Click to know more about EXPANDS">EXPANDS</a>, that is when <br/> `gammaP_(0) +V_(0) f.(V_(0)) gt 0` <br/> `f.(V_(0)) gt -gammaP_(0)//V_(0)`</body></html>


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