One level in which the contributions of Ernest Rutherford are important to the improvement of our lives is embodied in the process by which Rutherford worked. Rutherford embraced the idea that scientific progress is a collective entity, something in which individuals are able to find a communitarian sense of the good. The stereotypical notion of the scientist being isolated in their lab is something that Rutherford's contributions repudiate. Rutherford embraced science as part of a larger community. For example, he gained prominence by being a "leading figure in the scientific community, corresponding with scientists around the world, advising appointments in Europe, Australia, and North America, and playing a major part in the Royal Society." Rutherford viewed science as something collective, as seen in his role in making Manchester University a haven for scientists from around the world to share in collaboration on projects. This same sense of collective advocacy can be seen in Rutherford's work in World War I, where he was able to act as a point of convergence for naval officers and scientists. Rutherford's contributions demonstrate how being able to forge bonds and construct communities can result in great work products. This represents a point of improvement in our lives as we struggle to grasp the implications of an uncertain world and all that is a part of it. Rutherford's example can help guide us in recognizing that more is gained from worldwide collaboration as opposed to competition.
In a more practical sense, Rutherford's work in nuclear energy was pioneering. His thought and work makes him one of the critical forces of the atomic age. As early as the first part of the 1930s, Rutherford saw the potential energy which was untapped in the splitting of the atom:
We might in these processes obtain very much more energy than the proton supplied, but on the average we could not expect to obtain energy in this way. It was a very poor and inefficient way of producing energy, and anyone who looked for a source of power in the transformation of the atoms was talking moonshine. But the subject was scientifically interesting because it gave insight into the atoms.
It was clear to Rutherford that atomic energy could be envisioned. It could be something that might provide an unlimited energy source to human beings. The fact that the atomic age transformed how we view energy is a reason why Rutherford's contributions can impact our lives. One might be able to argue that Rutherford's articulation of atomic energy and the potential within it transformed human existence into something more complex and intricate. Rutherford was able to see how the power of the atom could transform what is into what can be. Given how atomic energy is now understood as a benchmark of science, our lives have become improved and advanced as a result of Rutherford's work.
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To summarize the main points of akannan's excellent and detailed answer as you ask for:
Rutherford changed the way we live by positing the idea of employing atomic energy. We now use this energy to generate electricity (via nuclear power plants), among other things.
Additionally, Rutherford in a way put a face to laboratory science. He showed how working as part of a team can yield better results than working alone. As akannan states on this page,
Rutherford viewed science as something collective, as seen in his role in making Manchester University a haven for scientists from around the world to share in collaboration on projects.
In these ways, Ernest Rutherford changed both science and the way we live now.