Final answer:
Manganese-51 is most likely to decay by positron emission due to its low neutron-to-proton ratio, which is indicative of a nucleus that is more likely to undergo this type of decay to attain stability (b).
Step-by-step explanation:
Among the given options, manganese-51 is most likely to decay by positron emission. This is because the neutron-to-proton (n:p) ratio is crucial in determining the stability of a nucleus. For a stable isotope, the n:p ratio falls within a certain range known as the band of stability. When the n:p ratio is too low, a nucleus is more likely to undergo positron emission as a means to increase this ratio by converting a proton into a neutron.
The n:p ratios for the isotopes in question are as follows: chromium-53 with a ratio of 28/24 = 1.21; manganese-51 with a ratio of 26/25 = 1.04; and iron-59 with a ratio of 33/26 = 1.27. Since Mn-51 has the lowest n:p ratio of these, it is more likely to experience positron emission, and it is noted that Cr-53 is a stable isotope while Fe-59 decays by beta emission.