# The entropy of a bath tub

according to the 2nd law of thermodynamics, as time has increase, disorder universally is increasing among the interaction of particles , eventually reaching thermo-equibrium.

As you probably know from experience, when you fill a tub with hot water, it doesn't take very long for the tub to become cold.
What I don't understand is , if a tub is filled with cold water, then why doesn't it become hotter as time passes bye? The hotter a system is, the higher the amount of disorder within that system. Of course if you leave a cold bottle of water inside a car for a certain amount of time, its no longer worth drinking after a 5 mile jog around the neighborhood. Why isn't the second law thermodynamics consistent for both systems?

marcus
Gold Member
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according to the 2nd law of thermodynamics, as time has increase, disorder universally is increasing among the interaction of particles , eventually reaching thermo-equibrium.

As you probably know from experience, when you fill a tub with hot water, it doesn't take very long for the tub to become cold.

to make the laws work, isolate the system

It does become hotter: The cold bath slowly warms up to room temperature. To get hotter than that, it would either have to: keep taking heat from the surroundings (which would decrease the entropy of the surroundings more than it would increase the entropy of the bath, and so is disallowed by the same 2nd law -- the "most probable state" is the one where everything is the same temperature) or else violate the 1st law (energy has to come from somewhere).

As you probably know from experience, when you fill a tub with hot water, it doesn't take very long for the tub to become cold.
What I don't understand is , if a tub is filled with cold water, then why doesn't it become hotter as time passes bye? The hotter a system is, the higher the amount of disorder within that system. Of course if you leave a cold bottle of water inside a car for a certain amount of time, its no longer worth drinking after a 5 mile jog around the neighborhood. Why isn't the second law thermodynamics consistent for both systems?

It will become "hotter" as time goes by; the cold tub water will slowly absorb heat from the air (which is assumed at room temperature) until the two are in thermal equilibrium with each other (the hot bath is the converse...it will cool down as it loses energy while it heats up the surrounding air until the water has dropped to room temperature). This might not be immediately obvious to you because water has a high heat capacity (means it takes quite a bit of energy to heat it up...especially when you are talking about a tub as opposed to say a glass of water). However, fill up a bowl of cold water and a bowl of hot water and come back a day later and check each one. They will be at the same temperature as the surrounding air.

In either case, energy is flowing from the hot body to the colder body so don't worry, thermodynamics is still safe

Andrew Mason
Homework Helper
The hotter a system is, the higher the amount of disorder within that system.
You have to be careful. It is confusing to use the concept of 'disorder' to explain entropy.

One might think that a very hot gas is more disordered than a cold gas and therefore has a higher entropy state. This is not correct.

I would suggest that you do not think of entropy as a measure of disorder. Rather, think of entropy as a measure of how dispersed the energy is. If a hot body and a cold body are in thermal contact, the concentrated energy in the hot body disperses throughout the two and becomes more disperse. Thus there is an increase in entropy.

If you cause the heat to reverse its flow, you must add work from the surroundings. In that way, the entropy of the system decreases -- but the entropy of the system + surroundings always increases. In other words, the energy in the surroundings changed from a more concentrated state to a more disperse state such that its increase in entropy exceeded the decrease in entropy of the system.

AM

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1) how many different ways can you rearrange the microstates of the system (the individual properties of each atom, ie. momentum, position) and still get the system to have the same macroscopic properties (temp., pressure, ect.)
2) count the number of digits in that huge number (or mathematically, just find logarithm of that number)
3) that is entropy, you can think of it as the amount of information that you don't know about the system (that is why digits are more important than the actual number... 3.14159 gives you twice as much info as 3.14)... entropy increasing is due to the fact that the complexity (amount of information in system) is constantly increasing

"if a tub is filled with cold water, then why doesn't it become hotter as time passes by"...

it does heat up, just more slowly... when a fast moving water molecule leaves the tub, the water cools down, when a fast moving water molecule comes back into the tub, the water heats back up... if you put a lid on the bathtub the water would heat up and cool down at the same rate... but because the room is so much bigger than the tub it gets complicated... the fast moving atom leaves the tub, cooling it down, but then it might go somewhere to the far side of the room instead of staying over the tub water and eventually coming back in returning its energy... so you end up with more fast moving atoms leaving the tub than coming back in.... this is why you blow on your tea to cool it down. Fast moving atoms naturally leave the tea, and hang out above the tea (this is why the air above the tea gets warmer), they might come back in, heating the tea up again, so you blow them away from the tea ensuring that they can't come back into the tea and return their energy... also because blowing into the tea helps those hot atoms that are ready to leave but are stuck underneath some cooler atoms get to the surface and escape. This is the way Feynman explains it in his lectures, which I would recommend if you enjoy those conceptual epiphanies... also, if you want a book that answers your question completely and goes over thermodynamics in a very excellent and interesting way... "Warmth Disperses and Time Passes: The History of Heat" by Hans Christian Von Baeyer

to answer that question less conceptually.... Entropy=Heat/Temperature.... the quantity heat over temperature aways increases... so if you increase the temp. of your system.. the heat must increase by a larger margin... heat is the energy contained in molecular motion, you can think of temperature as a sort of concentration (T=heat/entropy)... the total energy of the atoms in their particular arrangement, over the amount of particular arrangements that they could have been in.

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