Find the pressure `P` , in the diagram as shown, of monoatomic gas of one mole in the process `ABC` if `Delta U // Delta W= 6//7`

Options

A. `(5)/(2) P_(0)`
B. `(3)/(2) P_(0)`
C. `2 P_(0)`
D. `(7)/(2) P_(0)`

Answer

A

Explanation

In process `AB`,
Pressure = constant, `V prop T`
`V_(B) = 2V_(0)`
So, `T_(B) = 2 T_(0)` (if `T_(A) = T_(0)`)
`Delta W = P_(0) (2 V_(0) - V_(0)) - P_(0)V_(0) = RT_(0)`
`Delta Q = C_(P) Delta T = (5)/(2) RT_(0)`
`Delta U = Delta Q - Delta W = (3)/(2) RT_(0)`
In process `BC`,
Volume = constant `P prop T`
`P_(B) = P_(0)` but `T_(B) = 2 T_(0)`
So, `P_(c ) = KP_(0), T_(c ) = KT_(B) = K xx 2 T_(0)`
`Delta W = 0`
`Delta Q = Delta U = C_(v) Delta T`
` = (3)/(2) R xx 2T_(0) (K - 1) = 3 RT_(0) (K - 1)`
`Delta Q_(Total) = (5)/(2) RT_(0) + 3 RT_(0) (K - 1)`
`Delta U_(Total) = (3)/(2) RT_(0) + 3 RT_(0) (K - 1)`
`Delta U // Delta Q = 6//7`
Compare Eqs. (i), (ii) and (iii), `K = 5//2`
So, `P = (5)/(2) P_(0)`