$${\lambda _p} = {h \over {{P_p}}}$$

$${\lambda _e} = {h \over {{P_e}}}$$

$$\because$$ $${\lambda _p} = {\lambda _e}$$

$$ \Rightarrow {P_p} = {P_e}$$

$${(K)_p} = {{P_p^2} \over {2{m_p}}}$$

$${(K)_e} = {{P_e^2} \over {2{m_e}}}$$

K_p < K_e as m_p > m_e

Option (a)

A $$\to$$ B $$\to$$ C (stable)

Initially no. of atoms of B = 0 at t = 0, no. of atoms of B will starts increasing & reaches maximum value when rate of decay of B = rate of formation of B.

After that maximum value, no. of atoms will starts decreasing as growth & decay both are exponential functions, so best possible graph is (d).

$${N \over {{N_0}}} = {\left( {{1 \over 2}} \right)^{{t \over {{t^{1/2}}}}}}$$

$${N \over {{{10}^{10}}}} = {\left( {{1 \over 2}} \right)^{{{30} \over {60}}}}$$

$$ \Rightarrow N = {10^{10}} \times {\left( {{1 \over 2}} \right)^{{1 \over 2}}} = {{{{10}^{10}}} \over {\sqrt 2 }} \approx 7 \times {10^9}$$

A $$\to$$ B, B $$\to$$ C

$${{d{N_B}} \over {dt}} = \lambda {N_A} - \lambda {N_B}$$

$${{d{N_B}} \over {dt}} = 2\lambda {N_{{B_0}}}{e^{ - \lambda t}} - \lambda {N_B}$$

$${e^{ - \lambda t}}\left( {{{d{N_B}} \over {dt}} + \lambda {N_B}} \right) = 2\lambda {N_{{B_0}}}{e^{ - \lambda t}} \times {e^{\lambda t}}$$

$${d \over {dt}}({N_B}{e^{\lambda t}}) = 2\lambda {N_{{B_0}}}$$, on integrating

$${N_B}{e^{\lambda t}} = 2\lambda t{N_{{B_0}}} + {N_{{B_0}}}$$

$${N_B} = {N_{{B_0}}}[1 + 2\lambda t]{e^{ - \lambda t}}$$

$${{d{N_B}} \over {dt}} = 0$$ at $$ - \lambda [1 + 2\lambda t){e^{ - \lambda t}} + 2\lambda {e^{ - \lambda t}} = 0$$

$${N_{{B_{\max }}}}$$ at $$t = {1 \over {2\lambda }}$$