Which of the following statement(s) is (are) true? a. A radioactive nuclide that decays from \(2.00 \times 10^{21}\) atoms to $5.0 \times 10^{20}$ atoms in 16 minutes has a half-life of 8.0 minutes. b. Nuclides with large \(Z\) values are observed to be \(\alpha\) -particle producers. c. As \(Z\) increases, nuclides need a greater proton-to-neutron ratio for stability. d. Those light nuclides that have twice as many neutrons as protons are expected to be stable.

We are given the initial number of atoms (2.00 x 10^21), the final number of atoms (5.0 x 10^20), and the time elapsed (16 minutes). The formula for the decay of radioactive nuclides is: \[N(t) = N_0 e^{-\lambda t}\] where \(N(t)\) is the number of atoms at time \(t\), \(N_0\) is the initial number of atoms, \(\lambda\) is the decay constant, and \(t\) is the time elapsed. Let us find the value of the decay constant, \(\lambda\). Rearrange the equation and solve for \(\lambda\): \(e^{-\lambda t} = \frac{N(t)}{N_0}\) \(-\lambda t = \ln{\frac{N(t)}{N_0}}\) \(\lambda = -\frac{\ln{\frac{N(t)}{N_0}}}{t}\) Plug given values into the formula: \(\lambda = -\frac{\ln{\frac{5.0 \times 10^{20}}{2.00 \times 10^{21}}}}{16\:minutes}\) \(\lambda \approx 0.035 \: min^{-1}\) Now, we can use the decay constant \(\lambda\) to find the half-life (\(t_{1/2}\)) by using the formula: \(t_{1/2} = \frac{\ln 2}{\lambda}\) \(t_{1/2} = \frac{\ln 2}{0.035\: min^{-1}}\) \(t_{1/2} \approx 19.71 \:minutes\) Since the calculated half-life is not equal to 8.0 minutes, statement (a) is false.

Nuclei with large \(Z\) values (meaning a high number of protons) are more likely to emit α-particles to lower their overall size and energy. α-particle emission consists of the loss of 2 protons and 2 neutrons as a helium nucleus, reducing the \(Z\) value of the original nucleus. This is a well-documented phenomenon, so statement (b) is true.

As the number of protons (\(Z\) value) in a nucleus increases, the Coulomb repulsive force between them increases too. In order to maintain stability against this repulsive force, a greater number of neutrons are required. This implies that a larger proton-to-neutron ratio is needed for stability as \(Z\) increases. Therefore, statement (c) is true.

Generally, in light nucleus with Z < 20, stable nuclides tend to closely follow a 1:1 ratio between the number of protons and neutrons. If there are twice as many neutrons as protons in a given nucleus, it would not satisfy this criterion for stability. This excess in neutrons would typically result in instability due to β-decay. Therefore, statement (d) is false. In conclusion: Statements (b) and (c) are true, while statements (a) and (d) are false.