Natural Log in Thermodynamic Equations related to Entropy.

[eliassiguenza]

Well I've been doing fairly complex thermodynamics in classical mechanics lectures with a bit of medium to advanced class, first of all , it took me a very long time to wrap my head around the concept of Entropy, however I still don't understand why we use ln (Natural Log) to calculate the amount of entropy when a system has internal work (i.e the classical experiment of putting two gasses in an adiabatic process and letting them mix then having to calculate the entropy generated.) Also i can't really get a grip of the idea of maximum entropy. Is it a quantity? is it a concept only ? what is it ... ?

It was not until last nite that i found a lecture of Leonard Susskind that I truly started grasping the concept of entropy. Still that was related to Statistical Mechanics, and I'm not there yet, I am in Classical Physics.

I honestly would appreciate a good answer.
Thank You!

 

[rock.freak667]

Hello eliassiguenza,

I am not sure how entropy is explained in statistical thermodynamics but in classical thermodynamics, entropy is considered to be a measure of the energy that cannot be converted into useful work.

This site, http://wright.nasa.gov/airplane/entropy.html, shows the derivation of the formula and the natural log is as a result of integrating something of the form dx/x.

I don't think one can actually measure entropy, so it is a concept essentially.

From the second law of thermodynamics, in a closed system, the entropy of a system increases to a maximum value. So say you have an engine that received some heat energy Q and is rejecting heat Qc and thus is doing work W. (the system is open so the entropy does not necessarily increase)

If you stop the supply of Q, then the system becomes closed and the heat energy left for the engine to do work is limited, so the energy that can be converted to work decreases as the engine continues to use up that energy. When the engine stops producing work it is because the entropy has reached its maximum value and no more heat energy can be converted into work.

 

[Andrew Mason]

Well I've been doing fairly complex thermodynamics in classical mechanics lectures with a bit of medium to advanced class, first of all , it took me a very long time to wrap my head around the concept of Entropy, however I still don't understand why we use ln (Natural Log) to calculate the amount of entropy when a system has internal work (i.e the classical experiment of putting two gasses in an adiabatic process and letting them mix then having to calculate the entropy generated.)

To calculate the change in entropy you must use the reversible path between the beginning and end states. The beginning state is with the two gases each occupying 1/2 of the total volume. The end state is with each gas occupying the entire volume but at 1/2 the initial density and no change in temperature.

So the entropy calculation is for an isothermal expansion from V/2 to V for each of the gases:

$$\Delta S = \int dS = \int dQ/T = \int dU/T + \int PdV/T = 0 + \int PdV/T$$

Since P = nRT/V,

$$\Delta S = \int PdV/T = nR\int dV/V = nR\ln{(V_f/V_i)} = nR\ln(2)$$

Also i can't really get a grip of the idea of maximum entropy. Is it a quantity? is it a concept only ? what is it ... ?

If you are referring to Jaynes' maximum entropy principle, it is a difficult principle of statistical analysis that is best explained by someone who really understands it - not me.

AM

 

[eliassiguenza]

Thank you so much guys! you have cleared my mind specially , rock.freak667 your really easy explanation on the maximum entropy! god I reckon i hard to understand this concepts at 7.00 am .. :s , Andrew Mason thank you so much and yeah I know it's a crazy subject that maximum entropy but thank god I'm not doing statistical thermodynamics yet, soon tho! I really appreciated your time !