An ideal gas is expanding such that `PT^(2)= `constant. The coefficient of volume expansion of lthe gas is:

Options

A. `(1)/(T)`
B. `(2)/(T)`
C. `(3)/(T)`
D. `(4)/(T)`

Answer

C

Explanation

`PT^(2)=` constant (Given) ..(i)
According to ideal gas equation, `PV = nRT or P = (nRT)/(V)`
Substituting this value of P in (i), we get
`((nRT)/(V)) T^(2)=` constant or `(T^(3))/(V)` = another constant
Differentiating both sides, we get
`(1)/(V) 3T^(2) dT - (T^(3))/(V^(2)) dV= 0`
or `3dT = (T)/(V) dV or (1)/(V) ((dV)/(dT)) = (3)/(T)` ...(ii)
According to definition of the coefficient of volumne expansion of the gas is
`gamma = (1)/(V) ((dV)/(dT))`....(iii)
From (ii) and (iii), we get `gamma (3)/(T)`