Derive an integrated rate equation in terms of total pressure (P) and the partial pressures `P _(A), P_(B), P_(C)` for the gaseous reaction `A (g) to B(g) +C(g).`

Explanation

Given first order gas phase equations
`A(g) to B(g) +C(g)`
Let `p_(i)` be the initial pressure of A and `p_(t)` the total pressure at time 't'. Integrated rate equation for such the reaction can be derived as
Total pressure `P_(t)=P_(A) +P_(B)+P_(C)` (prssure units)
`P_(A), P_(B) and P_(C)` are the partial pressure of A, B and C, respectively.
If x atm be the decrease in pressure of A at time t and one mole each of B and C is being formed, the increase in presure of B and C will also be x atm each.
`{:(A(g),to, B(g), +, C(g)),(At t=0p_(i) atm, ,0 atm,,0atm),("At time"(p_(i)-x) atm, ,0atm,, 0atm):}`
where `p_(i)` is the initial pressure at time `t=0.`
`P_(t)=(P_(i)=x)+x+x=P_(i)+x`
`x= (P_(t)-P_(i))`
where `P_(A)=P_(i) =P_(i)-(P_(t)-P_(i))=2P_(i)-P_(t)`
`k=((2.303)/(t))(log ""(p_(i))/(p_(A)))`