Find the minimum attainable pressure of one mole of an ideal gas, if during its expansion its temperature and volume are related as `T=T_(0)+alphaV^(2)` where `T_(0)andalpha` are positive constants.

Explanation

Given that one mole of gas is used, thus from gas law, we have PV= RT or
`P=(RT)/V=R/V(T_(0)+alphaV^(2))" "["as "T=T_(0)+alphaV^(2)]`
Here pressure P will be minimum, when
`(dP)/(dV)=0or(dP)/(dV)=(RT)/V^(2)+alphaR=0orV=sqrt(T_(0)/alpha)`
Thus pressure of the gas is minimum, when its volume is
`V=sqrt(T_(0)/alpha)` and at this volume, its temperature is given as
`T=T_(0)+alphaV^(2)=T_(0)+alpha(sqrt(T_(0)/alpha))^(2)=2T_(0)`
Thus minimum value of pressure is
`P_(min)=(RT)/V=(R(2T_(0)))/(sqrt(T_(0)//alpha))=2Rsqrt(T_(0)alpha)`