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Beta (\(\beta\)) emission is a type of radioactive decay where a beta particle, which can either be an electron or a positron, is emitted from an unstable nucleus. This process occurs when there is an imbalance in the number of protons and neutrons in a nucleus.- In beta-minus decay, a neutron converts into a proton, and an electron (beta particle) and an antineutrino are emitted.- In beta-plus decay, a proton is transformed into a neutron, releasing a positron (beta particle) and a neutrino.Beta decay changes the identity of an element by altering its atomic number without changing its mass number. It often occurs in nuclei where the ratio of neutrons to protons is too high or too low, causing instability. For example, carbon-14 decays into nitrogen-14 through beta-minus decay, increasing the atomic number while keeping the mass number the same.Understanding beta emission is crucial in fields such as nuclear medicine, where radioactive isotopes are used for imaging and treatments.

The half-life of a radioactive substance is the time it takes for half of the radioactive nuclei in a sample to decay. This measure is crucial for understanding how quickly a radioactive material becomes stable by losing its instability over time.The half-life can be calculated using the formula:\[ T_{1/2} = \frac{\ln(2)}{\lambda} \]where \(\lambda\) is the decay constant and \(\ln(2)\) is the natural logarithm of 2, approximately 0.693.From the given problem, applying the process explained above, we calculate that the sample's half-life is 2 seconds. This means every 2 seconds, half of the remaining radioactive nuclei decay. Such calculations are widely used in fields like archaeology, to date organic materials, and in medicine, ensuring safe dosage levels of radioactive substances.

Radioactive decay is generally described by an exponential decay formula, which represents how the quantity of a radioactive substance decreases over time.The formula is:\[ N(t) = N_0 e^{-\lambda t} \]where:- \(N(t)\) is the number of radioactive atoms remaining at time \(t\).- \(N_0\) is the initial number of atoms.- \(\lambda\) is the decay constant.- \(e\) is the base of the natural logarithm, approximately equal to 2.718.This equation models how radioactive elements decay predictably over time. For example, in the given exercise, we can see how the amount reduces through an exponential function, establishing the predictable nature of decay as measured by the half-life. Exponential decay is not just limited to radioactive decay but is also applicable in fields like pharmacology and finance, where processes diminish over time.