InterviewSolution
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InterviewSolution
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Consider a `P-V` diagram in which the path followed by one mole of perfect gas in a cyclinderical container is shown in (figure) (a) Find the work done when the gas is taken from state 1 to state 2. (b) What is the ratio of temperatures `T_(1)//T_(2), if V_(2)= 2V_(1)`? (c ) Given the internal energy for one mole of gas at temperature `T is (3//2) RT`, find the heat supplied to the gas when it is taken from state 1 to 2, with `V_(2)= 2V_(1)`. |
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Answer» Let `pV^(1//2)` = Constant = `K, p=(K)/(sqrt(V))` (a) Work done for the process 1 to 2, ltBrgt `W=int_(V_(1))^(V_(2))pdV=Kint_(V_(1))^(V_(2))(dV)/(sqrt(V))=K[(sqrt(V))/(1//2)]_(V_(1))^(V_(2))=2K(sqrt(V_(2))-sqrt(V_(1)))` `=2p_(1)V_(1)^(1//2)(sqrt(V_(2))-sqrt(V_(1)))=2p_(2)V_(2)^(1//2)(sqrtV_(2)-sqrt(V_(1)))` (b) From ideal gas equation, `pV=nRTrArrT=(pV)/(nR)=(psqrt(V)sqrt(V))/(nR)` `rArr" "T=(Ksqrt(V))/(nR)" "("As",psqrt(V)=K)` Hence, `" "T_(1)=(Ksqrt(V_(1)))/(nR)rArrT_(2)=(Ksqrt(V_(2)))/(nR)` `rArr" "(T_(1))/(T_(2))=((Ksqrt(V_(1)))/(nR))/((Ksqrt(V_(2)))/(nR))=sqrt(V_(1)/(V_(2)))=sqrt(V_(1)/(2V_(1)))=(1)/(sqrt(2))" "(because V_(2)=2V_(1))` (c ) Given, internal energy of the gas = `U=((3)/(2))RT` `" "DeltaU=U_(2)-U_(1)=(3)/(2)R(T_(2)-T_(1))` `" "=(3)/(2)RT_(1)(sqrt(V)-1)" "[becauseT_(2)=sqrt(2)T_(1)"from"(b)]` ltbgt `" "DeltaW=2p_(1)V_(1)^(1//2)(sqrt(V_(2))-sqrt(V_(1)))` ltBrgt `" "=2p_(1)V_(1)^(1//2)(sqrt(2)xxsqrt(V_(1))-sqrt(V_(1)))` `" "=2p_(1)V_(1)(sqrt(2)-1)=2RT_(1)(sqrt(2)-1)` `because " " DeltaQ=DeltaU+DeltaW` `" "=(3)/(2)RT_(1)(sqrt(2)-1)+2RT_(1)(sqrt(2)-1)` `" "=(sqrt(2)-1)RT_(1)(2+3//2)` `" "=((7)/(2))RT_(1)(sqrt(2)-1)` This is the amount of heat supplied. |
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