What appeal to emotion does Jefferson use in the first paragraph of the Declaration of Independence?
Limiting the answer to the first paragraph alone, Jefferson's use of the phrase "a decent respect to the opinions of mankind" can be construed as an appeal to emotion.
Strictly speaking, the Declaration of Independence could have been just that: a plain legal assertion that the American colonies represented by the Declaration's signatories were no longer under English rule.
Instead, by asserting that "a decent respect to the opinions of mankind" requires that the text enumerate the complaints of the colonists, Jefferson employs an "appeal to consequences," a subset of emotional appeal. In effect, he's saying that unless the Declaration details exactly why the Colonies are declaring independence from England, they risk being deemed unjust or unreasonable by the rest of the world.
That appeal sets up the rest of the Declaration. The second paragraph of the Declaration tends to be the part Americans remember. It's the bit with the beautiful assertions of "all men are created equal" and "life, liberty and the pursuit of happiness."
The rest of the Declaration is rather different. It consists in large part, to use its own language, of "a long train of abuses and usurpations" that the authors attributed to the English crown. By appealing to consequences in the first paragraph, Jefferson and the Declaration's other contributors give themselves license to list various terrible things they felt were inflicted on them by English authority. In doing so, they hoped to cast themselves as the injured party in the conflict and justify their actions on the global stage.
To state the obvious, it worked. America won the War of Independence, in large part due to foreign aid, particularly from France. By all appearances, Jefferson's appeal to emotion did its job admirably.
In the first paragraph, Jefferson references the "decent respect to the opinions of mankind" that require the colonies to "declare the causes which impel them" to sever the political bonds that once tied them to England. In this way, then, he attempts to call on not only his readers' sense of logic—that such a separation is reasonable and therefore right—but also their emotions. He offers the idea that, out of respect for humanity, he feels an obligation to explain the reasons for this separation, even contextualizing this particular situation between England and the colonies within the "Course of human events." This reads, to me, like Jefferson is trying to appeal to the humanity of his readers, an emotional appeal.
Further, he also uses the word "entitle" which can be read both logically and emotionally. According to him, both Nature and God "entitle" the colonies to a separate status equal to England's, and this entitlement is granted them by their superior logic and heightened awareness of what is right and good for humanity.
In the first paragraph of the Declaration of Independence, Jefferson appeals to the "Laws of Nature," and "Nature's God" in order to make his argument for separation. He states that the colonists are seeking "the separate and equal station" to which the colonists are entitled to have. Jefferson was not a Christian in the truest sense of the word, but his religious preferences can best be described as Deism. Jefferson was heavily influenced by the English philosopher John Locke who believed that nature had God-given natural laws. Jefferson used the argument that it was God's will for the colonies to seek separate status at this time, as the colonies and Britain could no longer be reconciled. Another reason Jefferson used this language was to capture religious sentiment in the colonies--if given the holy cause of creating a new nation, the colonists would try harder to do the nearly impossible task of leaving Britain.
Jefferson largely appeals to logic, not emotion, in the opening paragraph of the Declaration of Independence. The phrase "it becomes necessary" in the first sentence of the declaration implies there are certain criteria that, when met, justify breaking free from an existing country. Jefferson then cites "equal station" that "laws of nature" provide, which can be interpreted as an objective measure of inherent rights into which people are born. Finally, Jefferson notes that making a legitimate case for independence requires respecting "the opinions of mankind" by providing a clear explanation for why independence is necessary.
While the question is about the first paragraph, it is worth considering the second paragraph since the primary function of the former is to set the stage for the latter. The second paragraph contains some of the most famous phrases of the Declaration of Independence, expressing the right and obligation of a people to break free from an oppressive government "destructive to these ends." According to Jefferson, when serious grievances are left unaddressed people have the right to seek independence so long as they show how their basic rights were infringed.
In short, the first paragraph appeals mostly to logic. Jefferson's language is measured and points to natural law in an effort to convey objectivity through statements of fact and reason. He states that a cause for independence can be warranted if properly explained, and considered legitimate if grounded in just causes. What follows in the remainder of the Declaration of Independence are the supporting points to prove his assertion.