As scores of angry depositors appealed for their money, the Argentine government yesterday shut down the country's fifth-largest private bank and ordered its liquidation because of financial problems.

The order to liquidate the Banco de Italia y Rio de la Plata (Bank of Italy and the River Plate) was disclosed in a communique' issued by the country's Central Bank in the late afternoon. The bank had been shut down by the government at the start of the business day.

The communique' cited the "grave situation that the entity presented in terms of solvency and liquidity." It said the Central Bank had recently ordered the private bank to come up with sufficient capital to carry out a plan for correcting financial problems.

"These funds were not provided in the time or amount needed for the Banco de Italia y Rio de la Plata to return to carrying out its activities in a normal and reasonable manner," the communique' said.

The private bank, with 88 branches and about 3,000 employes, had 18.6 billion pesos in deposits in October, the last month for which figures were available. At the exchange rate then in effect, the total equaled about $132.6 million, and was the fifth-highest total for a private bank and 11th-highest for both private and state banks. Previous public statements show a steady decline in the bank's assets. In May 1984, it had an estimated $250 million in deposits.