The half-life of a radioactive substance is the time it takes for half of the atoms in a sample to decay.
This decay process is inherent to the specific nucleus of the atom, meaning the half-life is a property of the substance itself and is independent of factors like the amount present or how old the sample is.
In beta-minus decay, a neutron is converted into a proton, increasing the number of protons and a corresponding increase in the atomic number of the element, while the mass number remains unchanged.
Thus, the new nucleus has one more proton than the original.