$$_{92}^{238}U$$ is known to undergo radioactive decay to form $$_{82}^{206}Pb$$ by emitting alpha and beta particles. A rock initially contained 68 $$ \times $$ 10^$$-$$6 g of $$_{92}^{238}U$$. If the number of alpha particles that it would emit during its radioactive decay of $$_{92}^{238}U$$ to $$_{82}^{206}Pb$$ in three half-lives is Z $$ \times $$ 10¹⁸, then what is the value of Z?

The decomposition reaction

$$2{N_2}{O_5}(g)\buildrel \Delta \over \longrightarrow 2{N_2}{O_4}(g) + {O_2}(g)$$

is started in a closed cylinder under isothermal isochoric condition at an initial pressure of 1 atm. After Y $$ \times $$ 10³ s, the pressure inside the cylinder is found to be 1.45 atm. If the rate constant of the reaction is 5 $$ \times $$ 10^-4s^-1, assuming ideal gas behaviour, the value of Y is ...............

Consider the kinetic data given in the following table for the reaction A + B + C $$ \to $$ Product.


The rate of the reaction for [A] = 0.15 mol dm^-3, [B] = 0.25 mol dm^-3 and [C] = 0.15 mol dm^-3 is found to be Y $$ \times $$ 10^-5 mol dm^-3s^-1. The value of Y is .................

Consider the following reversible reaction, $$A\left( g \right) + B\left( g \right) \to AB\left( g \right).$$

The activation energy of the backward reaction exceeds that of the forward reaction by $$2RT$$ (in $$J\,mo{l^{ - 1}}$$). If the pre-exponential factor of the forward reaction is $$4$$ times that of the reverse reaction, the absolute value of $$\Delta {G^ \circ }$$ (in $$J\,mo{l^{ - 1}}$$ ) for the reaction at $$300$$ $$K$$ is ____________.

(Given; $$\ln \left( 2 \right) = 0.7,RT = 2500$$ $$J\,mo{l^{ - 1}}$$ at $$300$$ $$K$$ and $$G$$ is the Gibbs energy)