The given reaction is A(g)→B(g)+C(g)
At t=0, the initial pressure of A is pi and the pressures of B and C are 0.
At time t, let the decrease in pressure of A be x.
Pressure of A =pi−x
Pressure of B =x
Pressure of C =x
Total pressure at time t is pt=(pi−x)+x+x=pi+x
From this, we get x=pt−pi
The partial pressure of A at time t is PA=pi−x=pi−(pt−pi)=2pi−pt
For a first order reaction, the rate constant k is given by
k=t1lnPAPA,0
Substituting the values, we get
k=t1ln2pi−ptpi
Answer: t1ln2pi−ptpi