A one mole of ideal gas goes through a process in which pressure p varies with volume V as `p =3-g((V)/(V_(0)))^(2)`, where, `V_(0)` is a constant. The maximum attainable temperature by the ideal gas during this process is
(all quantities are is SI units and Ris gas constant)

Options

A. `(2V_(0))/(3R)`
B. `(2V_(0))/( R )`
C. `(3V_(0))/(2R)`
D. None of these

Answer

D

Explanation

(No option is matching)
Using `pV=RT" and "p=3-g((V^(2))/(V_(0)^(2)))`, we get
`T=(3V)/( R )-("g V"^(3))/(RV_(0)^(2))`
Differentiating w.r.t. volume, we get
`(dT)/(dV)=(3)/( R )-(g)/(RV_(0)^(2))xx3V^(2)`
For maximum `T,(dT)/(dV)=0`
`implies(3)/( R )-(g)/(RV_(0)^(2))xx3V^(2)=0impliesV=(V_(0))/(sqrt(g))`
Substituting in Eq. (i), we get
`T_("max")=(3V_(0))/(Rsqrt(g))-(g(V_(0)//sqrt(g))^(3))/(RV_(0)^(2))`
`=(2V_(0))/(Rsqrt(g))`.