Final answer:
Hermann von Helmholtz gained Ludwig Mond's approval and admiration through his contributions to physics, notably the theory of the Sun's energy production and his intellectual integrity. His holistic approach to science, which included the humanities, further enhanced his reputation among peers.
Step-by-step explanation:
Hermann von Helmholtz gained Ludwig Mond's approval and admiration through his significant contributions to science, particularly in the field of physics. Helmholtz's proposal, along with British physicist William Thomson (Lord Kelvin), that the Sun's energy could be explained by the conversion of gravitational energy into heat, suggested an innovative mechanism by which the Sun's outer layers might be 'falling' inward due to gravity. This process implied that the Sun could be shrinking in size while maintaining its heat and luminosity. Helmholtz's intellectual prowess, along with his personal integrity and influence on colleagues and students, especially during the horrors of Nazi oppression, solidified his prominent place in history and certainly accrued him immense respect and accolades from peers including Mond.
In addition to his scientific achievements, Helmholtz's broad interests and his ability to intertwine the humanities with science further cemented his reputation. His holistic view of science, akin to the enjoyment one finds in music, art, and philosophy, was seen as enlightening and contributed to a well-rounded appreciation of his character and intellect.