Diplomat. Born in Edinburgh, Mitchell was the son of a minister of St. Giles Kirk who one of the King's Chaplains in Scotland. He married young, but both his wife and their infant daughter died soon after, leaving Mitchell devastated but with substantial estates in Aberdeenshire. He left Scotland, first travelling in Europe and then studying law in England.
He was elected a Fellow of the Royal Society in 1736 and, in 1742, was appointed Under-Secretary of State for Scotland under John Hay, 4th Marquis of Tweeddale (1695 - 1762). He served in this role during the Jacobite Rebellion of 1745 and Mitchell corresponded with Duncan Forbes of Culloden (1685 - 1747) showing sympathy for the resulting suffering of both sides.
In 1756, Mitchell was sent as Ambassador to the Court of Frederick the Great of Prussia in Berlin. He negotiated the support of Prussia against the French during the Seven Years' War and wrote of his experiences in his journals. He visited England in 1765 and was knighted by King George II, before returning to Germany the following year.
He died in Berlin and was buried there.