Consider a P-V diagram in which the path followed by one mole of perfect gas in a cylindrical container as shown in Fig
a) Find the work done when the gas is taken from state 1 to state 2.
b) What is the ratio of temperature `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)` .

Explanation

a) Work done by the gas (Let` PV^(1//2)=A`)
`DeltaW=int_(V_(1))^(V_(2))pdv=A int_(V_(1))^(V_(2))(dV)/sqrtV=A[(sqrtV)/(1//2)]_(V_(1))^(V_(2))=2A(sqrtV_(2)-sqrtV_(1))=2P_(1)V_(1)^(1//2)[V_(2)^(1//2)-V_(1)^(1//2)]`
b) Since `T =pV//nR=A/(nR).sqrtV`
Thus `T_(2)/T_(1)=sqrt(V_(2)/V_(1))=sqrt2`
c) Then, the change in internal energy
`DeltaU=U_(2)-U_(1)=3/2R(T_(2)_T_(1))=3/2RT_(1)(sqrt2-1)`
`DeltaW=2AsqrtV_(1)(sqrt2-1)=2RT_(1)(sqrt2-1)`
`DeltaQ=(7//2)RT_(1)(sqrt2-1)`