Let X1,…,Xn be independent exponential random variables with respective hazard rates λ1,…,λn, and let Y1,…,Yn be independent exponential random variables with common hazard rate λ. This paper proves that X2:n, the second order statistic of X1,…,Xn, is larger than Y2:n, the second order statistic of Y1,…,Yn, in terms of the likelihood ratio order if and only if λ≥12n−1(2Λ1+Λ3−Λ1Λ2Λ12−Λ2) with Λk=∑i=1nλik,k=1,2,3. Also, it is shown that X2:n is smaller than Y2:n in terms of the likelihood ratio order if and only if λ≤∑i=1nλi−max1≤i≤nλin−1. These results form nice extensions of those on the hazard rate order in Paˇltaˇnea [E. Paˇltaˇnea, On the comparison in hazard rate ordering of fail-safe systems, Journal of Statistical Planning and Inference 138 (2008) 1993–1997].