Imagine I drop a sugar cube of 2 cubic centimeter into a cup of distilled water and then wait for the sugar molecule to break apart and dissolves into the soon to be solution. Is there any work done or diffusion don't require energy since otherwise life would not be possible in the first place?
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$\begingroup$ Notice that the spread of sugar molecules from an ordered crystal in one location to separate molecules throughout the fluid volume represents a significant increase in entropy. That has implications for work performed. $\endgroup$– Carl WitthoftCommented Feb 3, 2021 at 14:07
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1 Answer
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Diffusion does require work, but it requires a deeper analysis than that given in introductory mechanics - that of a single point particle with forces.
Simple mathematical analysis
In it simplest form diffudion can be seen as movement of a (relatively large) particle due to random collisions with the solvent molecules. One typically models it with the following Newton's equation (for simplicity I consider 1D case):
$$
m\dot{v} = -\gamma v +f(t),
$$
where $\gamma$ is the friction coefficient and $f(t)$ is a random force, which is typically taken to be delta-correlated Gaussian. One could now write down the expression for the isntantaneous power:
$$
P(t)=F(t)v(t) = - \gamma v^2 + vf(t),
$$
and integrate the first equation and the power to calculate the work done.
Sugar cube
The case with a sugar cube is a bit more complex, since the collisions of the molecules agains the cube first serve to transfer to the sugar crystal sufficient energy to break it into small pieces, which then diffuse through the water.
Life and energy
Returning to the mathematical discussion one could see diffusion as molecules transferring (a part of) their energy to the particle to accelerate it, and this energy partially being lost via dissipation (i.e., essentially returned to the molecules but in more "diffuse" form). This closely resembles the heat engine discussed in statistical physics and thermodynamics.
Life can be viewed as an example of such a heat engine: a live cell absorbes energy from the environment (in the form of sugars and other molecules), processes it to do some useful work (building itself and replicating), and ejecting the unused energy as a waste (digested products). In other words, life is a non-equilibrium process, one of the processes mediating the transformation of the driven out of equilibrium universe towards the state of maximum entropy, sometimes referred to as the thermal death.
