This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.

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Answer
10^{10}$ atoms
Here's how to solve the problem:
Given:
Part i: Calculate the number of disintegrations in the first second. This refers to the initial activity (), which is the number of disintegrations per second at the start.
Step 1: Convert the half-life to seconds.
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Here's how to solve the problem: Given: Initial number of atoms, N_0 = 10^10 atoms Half-life, T_1/2 = 30 days Number of remaining atoms for parts ii and iii, N_t = 10^4 atoms Part i: Calculate the number of disintegrations in the first second.
This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.