What is the maximum water temperature that a lake can have to retain a dissolved oxygen concentration of 0.262 mM?

I don't understand you mM abbreviation, but I am certain you can find it from the fresh water solubility charts here;

http://www.engineeringtoolbox.com/oxygen-solubility-water-d_841.html

To determine the maximum water temperature at which a lake can have a dissolved oxygen concentration of 0.262 mM, you would need to consider the solubility of oxygen in water, which is influenced by various factors including temperature, pressure, salinity, and the presence of other solutes.

One way to establish this temperature is by referring to the oxygen solubility chart. These charts typically show the relationship between temperature and the maximum amount of dissolved oxygen that water can hold at that temperature.

For the specific value of 0.262 mM, you would need to find the corresponding temperature on the chart. The chart will display different dissolved oxygen concentrations along the temperature axis. Locate the concentration closest to 0.262 mM and trace it to the corresponding temperature.

Keep in mind that oxygen solubility in water can vary slightly depending on factors such as pressure and salinity, so it's important to use a chart that accounts for these variables if they are applicable to the specific lake in question.