How and why did the ideas of an American empire change from the 1790s to the 1850s? Did the definition of "American Exceptionalism" change as well?
First of all, I should note that this question is clearly taken directly from a class assignment. Your instructor is probably expecting an answer that makes reference to various readings from your specific class. This might mean that any answer we can give will not be appropriate for your class since we do not know what readings you have examined or what your instructor has said in class. Without being able to make reference to readings or lecture notes, I would argue that ideas of American empire became much more expansive in the time frame that you mention. This was largely because of the fact that the US was becoming stronger. As America became stronger and more interested in empire, the idea of American exceptionalism became more aggressive and more broadly based.
In the 1790s, dreams of an American empire would have been very limited. The citizens of the new nation were aware that they were not a major world power. At that point, the territory of the US only extended as far as the Mississippi River. The US did not control that river and it did not even have any real control over much of its territory. This was still more than a decade before the War of 1812 and there were still British forts on American soil. White Americans were not present in large numbers in most of the land west of the Appalachians. If Americans thought of empire at that point, they would have thought about expanding to fill the territory that already belonged to them. They would not have been thinking much about the possibility of dominating most of the continent.
By the 1850s, this had changed. The three clearest causes of the change were the Louisiana Purchase, the industrialization of the country, and the Mexican-American War. The Louisiana Purchase made the US feel more powerful because it greatly expanded the national territory and removed the possibility of having a foreign power controlling the Mississippi. The war with Mexico enlarged the country further and made Americans feel that they were powerful enough to dominate all of their neighbors. This feeling of power came about in part because of the country’s increasing economic power which had come about through industrialization. By the 1850s, American ideas of empire were dominated by the idea of Manifest Destiny. Americans thought that they deserved to control all of North American and, perhaps, even more.
Along with these changes came a change in the nature of ideas about American exceptionalism. In the 1790s, American exceptionalism was based only on the idea of freedom. Americans at that time felt that their country was the freest and most democratic in the world. By the 1850s, Americans felt exceptional and special in more ways. They felt that their country offered more economic opportunity than any other. They felt that they were racially superior to most other peoples. They felt that their religion and culture made them superior to others as well. In other words, ideas about liberty became mixed with the ideas of Manifest Destiny to create a more aggressive and expansionist idea of American exceptionalism
Ideas of an American empire and American imperialism in the 1790s focused on expansion into the west (though that was by no means the extent of the country's ambitions—many watched and waited for the collapse of the Spanish empire to the south). Jefferson's Louisiana Purchase secured a vast tract of land at a remarkably low price. The US desire for territory, however, could not be satisfied so easily; expansion to the Pacific coast became a priority, and when land could not be purchased or negotiated away, fighting became an option. The Mexican American War of 1846 secured Texas for America, ushering in the Age of American Imperialism where expansion (now both on and off the continent into the Pacific) became the priority.
American Exceptionalism has long been a part of America's perception of itself. The early Massachusetts colonists set out to found "a city on a hill" (a phrase rather loftily intended to echo the Sermon on the Mount) that would be an example for all future civilizations. Revolutionary Thomas Paine saw America as one of the only true bastions of liberty. And Alexis de Tocqueville most famously gave birth to the term "exceptionalism" in 1831 when he described the country as "quite exceptional" in its pursuit of and path toward liberty, democracy, and equality. It was around this time also that the notion of Manifest Destiny began to circulate; many came to believe it was Americans' God-given mission to expand across the continent, bringing civilization, Christianity, and prosperity in their wake.
In short, between 1790 and 1850 ideas of American Empire grew rapidly, parallel to an equally rapidly increasing belief in American Exceptionalism; American Exceptionalism became both the motive and the rationale for expansion.
The ideals of American Empire changed drastically from the 1790s to the 1850s due to several factors. As a newly independent nation in the 1790s, America was just coming to terms with its lack of foreign supervision and coming into its own. As decades passed, the new government took over and started taking control of the nation, but this was not met very favorably by a great many people. This was the beginning of the Civil War that almost tore the country into two parts. A decisive result ensured that North and South were united and slavery was abolished, among other highlights of the war. And America emerged as a Democratic republic.
The idea of American Exceptionalism evolved during this time period and spread through the populace that America is a country of hope and is different from the rest of world in terms of its democracy and opportunities. This also lead to the industrial revolution and the great american hope.