The pair correlation function and the correlation energy density of two-dimensional electron liquids are calculated on the basis of the Salpeter's Debye-chain type expansion. When the two-dimensional plasma parameter (£=2πne4T-2) is small, the expansion to the first order gives the correlation energy to the order of (£21n£). It is shown that experimental results in the domain (£<0.5) can be reproduced by taking into account the first and the second order terms in this expansion.