A heat-conducting piston can freely move inside a closed thermally insulated cylinder with an ideal gas. In equilibrium the piston divides the cylinder into two equal parts, the gas temperature being equal to `T_0`. The piston is slowly displaced. Find the gas temperature as a function of the ratio `eta` of the volumes of the greater and smaller sections. The adiabatic exponent of the gas is equal to `gamma`.

Answer

`T_0[(n+1)^(2)/(4n)]^(gamma-1)/(2)`

Explanation

Let d be the distance through Which the piston has to be moved to male the volume of one part n times that of the other part. Then
n `=(V_(0)+Ad)/(V_(0)-Ad)` or `d =(n-1)/(n+1)xx(V_(0))/(A)`
Now consider any displacement of the piston by x. Since the pistionis conducting and the process of compressionis slow, the temperature of gases on the two sides of the piston will be the same."
`therefore (p_(0)v_(0))/(T_(0))=(p_(1)(V_(0)+Ax))/(T)=(P_(2)(V_(0)-Ax))/(T)`
Since it is a quasi static process
`p_(1)A+F_("agent")=p_(2)A or F_("agent")=(P_(2)-P_(1))A`
Then dW, elementary work done by agent `= F_(agent)dx=(P_(2)-p_(1))Adx`
or `dW=(m)/(M)RT((1)/(V_(0)-Ax)-(1)/(V_(0)+Ax))Adx`
`=(m)/(M)RT(2A^(2)xdx)/(V_(0)^(2)-A^(2)x^(2))`
Now `dU=(2m)/(M)C_(v)dT`
By 1st law of thermodynamics `dQ=dU+dW`
Here `dQ= 0``therefore dU=-dW`
or `(2m)/(M)C_(v)dT=(m)/(M)RT(2A^(2)xdx)/(V_(0)^(2)-A^(2)x^(2))`
or `(R)/(gamma-1)dT=RT(A^(2)xdx)/(V_(0)^(2)-A^(2)x^(2))` or `(1)/(gamma-1)overset(T)underset(T_(0))int(dT)/(T)=A^(2)overset(d)underset(0)int(xdx)/(V_(0)^(2)-A^(2)x^(2))`
or`(1)/(gamma-1) In (T)/(T^(0)) = A^(2)overset(d)underset(0)int(xdx)/(V_(0)^(2)-A^(2)x^(2))`
Put`( V_(0)^(2)-A^(2)x^(2))=z`. Then-`(2A^(2)xdx)=dz`
`therefore int(xdx)/(V_(0)^(2)-A^(2)x^(2))=int-(1)/(2A^(2))(dz)/(z)=-(1)/(2A^(2))Inz=(1)/(2A^(2))In(V_(0)^(2)-A^(2)x^(2))`
`therefore (1)/(gamma-1)in(T)/(T_(0))=A^(2)xx-(1)/(2A^(2))[In(V_(0)^(2)-A^(2)x^(2))]_(0)^(d)`
`=(1)/(2)[In(V_(0)^(2)-A^(2)d^(d))-InV_(0)^(2)]`
`=-(1)/(2)[In(1-(A^(2))/(V_(0)^(2)).((n-1)^(2)V_(0)^(2))/((n+1)^(2)A^(2)))]=(1)/(2)In(4n)/(n+1)^(2)`
or `In(T)/(T_(0))=-(gamma-1)/(2)In(4n)/(n+1)^(2)`
or `T = T_(0)[(n+1)^(2)/(4n)]^((gamma-1)/(2))`