Hi Arkansas native here, I wrote a paper about this for my Antebellum history course a few years ago. I'll include some background for any other readers who may not be familiar with the situation as you are.

Background and the Convention

After the initial wave of secessions throughout the South in 1861, Virginia, Arkansas, Tennesee, and North Carolina were still undecided. Arkansas, as the other states did, held a vote to call a secession convention and elected delegates to serve as representatives of the people of Arkansas. Most of the representatives at the time were not in favor of leaving. They took a wait-and-see approach.

After the Battle of Fort Sumter, the question of leaving came up again, and this time many of the delegates had started to flip. When Lincoln ordered Arkansas soldiers to arms, that was essentially the final straw. Many representatives considered Lincoln's order to be coercive, and they refused to raise arms against the southerners who had seceded.

When the vote occurred it was 64-5. Almost all of the elected unionists had flipped and David Walker (President of the Convention, prominent Arkansas politician) asked that the remaining delegates amend their votes, as the measure would pass anyway, and a unanimous decision would show Arkansas unity. Four of the representatives agreed to switch but one delegate from Fayetteville did not. His name was Isaac Murphy, a senator from Huntsville.

For Murphy, the people of Fayetteville had elected him on the promise that he would uphold their anti-secession interests at the convention. He did not intend to betray the people who had entrusted him to represent their interests. It's also worth mentioning that Murphy was not an Arkansas native, he was a Northern man from Pennsylvania and held anti-slavery beliefs, which likely influenced his anti-secession platform. So for these reasons despite enormous pressure he refused to yield and switch his vote. He also refused to sign the new state constitution that was adopted during the convention. After the convention and during the war, he stayed in Arkansas but received death threats

After the war, Murphy would be chosen as the first Governor of Arkansas during the Reconstruction period. Hopefully that answers your question, it does honestly boil down to he was very principled in his pro-union anti-slavery beliefs, and he didn't want to let down the people who voted for him to represent them for expressly that reason.

Further Reading and Sources

For more reading I'd recommend a book called Call the Roll (I'd write a full citation for it, but I unfortunately can't remember the author's name at the moment, I think my father currently has the book, I'll try to update this post tomorrow evening when I can get a hold of it). It's a little old but it has a ton of information about the secession convention and the people involved, in particular Isaac Murphy.

Christ, Mark (ed.). The Die Is Cast: Arkansas Goes to War, 1861. Butler Center for Arkansas Studies. 2010

Dougan, Michael. Secession Convention. Encyclopedia of Arkansas. Updated December 3, 2018.

Ibid. Isaac Murphy (1799-1882). Encylopedia of Arkansas. Updated February 11, 2019.

(Edit: Accidentally left out some brief personal information about Isaac Murphy)