# A thermos bottle containing milk is shaken vigorously

I have a conceptual physical chemistry question here.

A thermos bottle containing milk is shaken vigorously. Consider the milk as the system. Will the temperature rise as a result of the shaking?

My answer to this is yes, but I'm having trouble finding an equation that explains this result.

The best i have is the molecular kinetic energy formula that $$\mathrm{ke_{avg}= \frac{1}{2}mv^2=\frac{3}{2}kT}$$ So a change in kinetic energy is $$\mathrm{\Delta ke = \frac{3}{2} k \Delta T}$$

Is this the right approach?

## 2 Answers

A conceptual approach might work better, since the shaking system is not going to convert mechanical energy into thermal energy as neatly as something like Joule's Paddle Wheel. As the bottle is shaken, the liquid in it is swirled around into more or less randomly oriented vortices. These cancel one another out and slow to rest via friction with their container, and the energy they contained has to go somewhere. As you wrote, it winds up increasing the average velocity, and thereby temperature, of the particles in the milk.

This basic mechanical to thermal energy conversion is the source of the conversion factor from joules to calories. Joules are used to measure mechanical energy and calories were used to measure thermal energy. It was the paddle wheel experiment I linked above that first related the two to one another.

The challenge presented by trying to relate the work done on the bottle of milk to the thermal energy that winds up going into the milk is less mathematical than practical. On any given shake we would first put energy into the bottle (move it upward, let's say) then take energy back out (slow it to a stop) before putting energy back into the bottle (moving back downward) and taking energy out again (slowing to a stop a second time.) This repeats until we get tired of shaking, and if we measured the difference in all of the energy inputs and outputs that ought to equal the thermal energy added to the milk, but actually doing that for real would be a pain.

• I don't think this picture is quite right--I think you might be conflating the kinetic energy of the bottle as a whole with that of its contents. Yes, when you accelerate it upward you are adding KE to the bottle as a whole, and when you decelerate it you are taking KE away from the bottle as a whole. But when it comes to warming, what matters is the effect of shaking on the KE of the contents. And for this, when you suddenly acclerate the bottle, you add KE to its contents (by swirling, etc., as you describe). But when you suddenly reverse the direction of shaking by applying acceleration... Apr 28, 2021 at 5:18
• ...in the opposite direction, you are further adding KE to the contents. I.e., while acceleration and deceleration increase and decrease the KE of the bottle as a whole (by which I mean 1/2 mv^2, where m is the mass of the bottle, and v is the velocity relative to your reference frame), respectively, acceleration and deceleration both increase the KE of the contents. The analogy would be gas contained in a cylinder that has pistons at both ends. Compressing either piston in the same way equally adds to the KE of the gas, irrespective of the direction in which the work is being done. Apr 28, 2021 at 5:23

The temperature of the milk increases because of work done on the milk molecules which cause , increase the velocities of the moecules and since velocity is directly proportional to the temperature. Velocity α Temperature So the temperature of milk increases due to the shaking process. Although this effect is very small but heat up the molecules.