Which of the following statement(s) is(are) true? a. A radioactive nuclide that decays from \(1.00 \times 10^{10}\) atoms to \(2.5 \times 10^{9}\) atoms in 10 minutes has a half-life of 5.0 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.

: Given that a radioactive nuclide decays from \(1.00 \times 10^{10}\) atoms to \(2.5 \times 10^{9}\) atoms in 10 minutes. Let's calculate the half-life (\(t_{1/2}\)) using the decay equation and given information: Decay equation: \(N = N_0 e^{-\lambda t}\), where \(N\) is the final number of atoms, \(N_0\) is the initial number of atoms, \(\lambda\) is the decay constant, and \(t\) is time elapsed. First, find the decay constant using the given number of initial and final atoms and time elapsed. Rearranging the equation to find the decay constant: \(\lambda = \frac{\ln(N_0/N)}{t}\). Now, substituting given values: \(N_0 = 1.00\times10^{10}\) atoms, \(N = 2.5\times10^9\) atoms, and \(t = 10 \: \text{minutes}\). Calculating the decay constant: \(\lambda = \frac{\ln(1.00\times10^{10}/2.5\times10^9)}{10}\). Now, we will use the relation between the decay constant and half-life: \(t_{1/2} = \frac{\ln(2)}{\lambda}\). Finally, calculating the half-life: \(t_{1/2} = \frac{\ln(2)}{\lambda}\). Compare this value with the given half-life of 5.0 minutes.

: Large \(Z\) values indicate that the element has a greater number of protons. Elements with large \(Z\) values are more likely to emit \(\alpha\)-particles to reduce the Coulomb repulsion between protons and become more stable. Thus, this statement is true.

: As \(Z\) increases, the number of protons increases. Due to electrostatic repulsion, the nucleus becomes unstable. Hence, a greater proton-to-neutron ratio is required for stability as observing a higher number of neutrons reduces electrostatic repulsion and increases nuclear attraction. Thus, this statement is true.

: For light nuclides (less number of nucleons), the most stable isotopes have a proton-to-neutron ratio close to 1:1. If a light nuclide has twice as many neutrons as protons, the proton-to-neutron ratio will be 1:2. This nuclide is not expected to be stable because of the imbalance between protons and neutrons. Thus, this statement is false.