If in a certain process one substance turns into another, and the rate of generation of the product is proportional to the available quantity of the original substance, then such a phenomenon is called a process (or reaction) of first order. A substance whose initial quantity is m 0 turns into another substance and from the product thus obtained a second product begins to generate immediately. Both reactions are of first order; the proportionality factors are known; k 1 for the first process and k 2 for the second reaction. If x is the quantity of the first product generated during time t units y is the quantity of the second product.
(i) y at t = 2 is given by:
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Ans.
(i)
Sol. (common)
According to the given conditions
= k 1 (m 0 – x) and 
= k 2 (x – y)
From the first equation 
+ k 1 x = k 1 m 0 . This is a linear equation with integrating factor e k.t . So the solution is
given by xe k.t = k 1 m 0 
+ const. But for t = 0, x = 0 so const
= – m 0 . Hence x = m 0 (1 – 
). The equation (ii)
reduces to 
+ k 2 y = k 2 m 0 (1 – 
), which is again a linear equation. Hence
= k 2 m 0 
+ const.
But y = 0 at t = 0 so
const. = –m 0 + 
. Hence
y = m 0 + 
(k 1 
–k 2 
) ... (i)
At t = 2, y = m 0 + 
(k 1 
–k 2 
)
(ii)
Sol. 
y = m 0 .
(iii)
Sol. 
= m 0 + 
= m 0 + 
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