I believe this portrait is mis-identified. There have been Countesses Elizabeth of Salisbury (seen here and here), but the title was upgraded to Marquess/Marchioness in 1789, when the 7th Earl became the first Marquess. His name was James Cecil and he was married to Lady Emily Mary Hill in 1773. His son James Brownlow became the second Marquess in 1823. According to his Wikipedia article, "Lord Salisbury was married twice. His first marriage was on 2 February 1821 to Frances Mary Gascoyne (c. 1806 – 15 October 1839)..." He later married Lady Mary Catherine Sackville-West.
So this could be a portrait of the new Marchioness of Salisbury, Frances Mary, who lent her name alongside her husband's to their six offspring. Her article in thepeerage is here.