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It seems as if the 'lowest energy' state for the Earth is to not have any mountains and that given enough time, the oceans would erode all land. Essentially it's an argument like entropy. And I mean in the far future, once tectonic activity has ceased.
Is this the case, and if so how long might it take? If it is not the case, why not?
Continents are made up of different, lighter materials than oceans and both continents and oceans "float" on the mantle. Continents are also thicker than oceans so erosion does not only need to take care of the ~5 km which is above sea level but it should also erode away the continental root (which can be several 10's of kilometers until the continental crust is actually slightly thinner than oceanic crusts are today.
However tectonics is not the only source of topography which can easily be seen if we look at Mars where a hotspot has created an enormous mountain on a tectonically inactive planet.
So if you want to end up with a topography free planet you need to shut down all possible mantle plumes too.
I would argue that plate tectonics is the major reason their is elevation on the surface of the earth. But let me be clear that the oceans are not the major eroders of land - running water on the land is (from rain and snow). You are correct to assume that as the earth cools, plate tectonics will subside and the rate of erosion will overcome the rate of uplift due to plate tectonics.
As for making estimates about how long it will take to do this, I would not wish to hazard a guess. Earth has been going strong for billions of years and it probably has a few billion more years left.
Your question is partially flawed because it is not the oceans that erode continents. Instead it is the collective erosion known as denudation, the sum of all processes inclusing weathering, slope processes and fluvial processes that reduce topography on the continents.
The level to which these processes tend will be the ocean level, identified as the base level. That said a few processes may erode to levels below the base level. Deep weathering could possibly reache below this level but probably only to tens of metres. Glacial processes can definitely erode to kilometers below sea level but this is localised to valleys (probably following fracture zones in the bedrock) forming fjords that may be eroded down to 1 to 1.5 km below sea level.
To add to this general condition, we must consider that sea level fluctuates depending on the distribution of sea basins realtive to continents. New ocean floor is relatively high elevation when compared to older ocean floor due to density differences. Water can also be accumulated on land during cold periods which can cause sea levels to vary. Hence the sea level is far from constant through time.
An implication of the question is what will happen in the future. I will assume the earths interior will cool with time chaning the continental drifting. With slowing drifting mountain building will decrease so that denudation will likely be dominating. A problem here is that we probably need to consider loss of atmosphere and other issues that we do not know much about. There are indictions from Mars that water has been more abundant in teh past yet the topography on Mars is significant because the erosional processes have lost their poser as loss of water may have been critical.
So what will happen in the futiure depends on how tectonic and atmospheric processes change in the distant future due to changes or development in both the earth's interior and the atmosphere in terms of water content (water being a critical componete in both weathering of rocks abnd transport of material in topgraphic terrain).
To conclude, the question is much larger than can first be thought and we need to consider large time frames in which many of existing processes and systems will drastically change. So to provide a time frame is very difficult and essentially pointless.
