Mobil Corp., the nation's second-largest oil company, and Bethlehem Steel Corp., the second-largest integrated steel producer in the United States, reported a gain in second-quarter net income over the same period of 1980. However, the 20th-ranked oil company, Cities Service Co., reported a 15 percent decline in second-quarter earnings, but said that refining and marketing operations were profitable after posting a loss in the first quarter. m

Mobil said it earned $700 million ($1.65 a share) up from $688 million ($1.62) a year earlier. Revenues rose to $16.6 billion from $15.4 billion.Higher profits on U.S. energy operations more than offset a decline in overseas earnings, Mobil said today.

Mobil reported it made $279 million on U.S. energy operations, up 13 percent from a year earlier, but said foreign energy earnings of $453 million were down 4 percent.

For the first six months of the year, Mobil earned $1.34 billion ($3.15), down from the 1980 figures of $1.53 billion ($3.61). Revenues were $33.7 billion, up from $30.7 billion.

Cities Service said its second-quarter earnings were $90.5 million ($1.09 a share), down from $107 million ($1.29) in the same period of 1980. Revenues rose to $2.27 billion from $1.84 billion.

The 1981 quarter included a profit of $13 million resulting from the settlement of helium contract litigation with the federal government.

Cities Service reported a six-month profit of $148.2 million ($1.78), down from $277.1 million ($3.33). Revenues were $4.56 billion, up from $4.05 billion.

Bethlehem Steel Corp. reported yesterday that net income in the second quarter increased 80 percent over the same period of 1980.

The income of $77.6 million for the period ended June 30, compared with $43.1 in the second quarter last year, came on net sales of $1.9 billion, up 18 1/2 percent over sales of $1.6 in the same period a year ago.

The second-quarter figures give the nation's second-largest steelmaker sales of $3.8 billion for the first six months this year, a 6 1/2 percent increase over the $3.57 billion on sales for the first six months of 1980.

Net income of $103.5 million for the first half was up 6.2 percent from the income during the same period last year, $97.5 million.