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MathError

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- Homework Statement
- Find entropy variation due to irreversibility when heat sources at two different temperatures (T_h>T_c) exchange heat (Q)

- Relevant Equations
- T_c=temperature of cold source

T_h=temperature of hot source

Q= exchanged heat

ΔS= entropy variation

ΔS_irr= entropy variation due to irreversibility

Hello to everyone, I'm studying thermodinamics and I would like to understand better the meaning of entropy and how to calculate it.

I know that if A and B are two possible states of a system, the equation whcih defines variation of entropy from A to B is:

ΔS=∫(dQ/T)_rev (1)

So I have to find a generic reversible transformation, which go from A to B, and resolve the integral along this transformation. In this case system would evolve through infinite sources which are step by step at the same temperature of system.

I have thought to this example where two heat sources at different temperatures (T_h>T_c) exchange an amount of heat (Q).

As reversible transformations I have choosed isotermal transformations at the two different temperatures of the sources (you can see my solution in the pic below).

Now I start to have some problems. Let's consider the equation (1), that integral can be calculated also on the real (irreversible) transformation which system follows and we have that:

ΔS=∫(dQ/T)_rev≥∫(dQ/T)_irr (2)

In general it can be written:

∫(dQ/T)_rev=∫(dQ/T)_irr+ΔS_irr

where the term ΔS_irr is a contribution due to irreversible process.

What I have tried to do is to calculate ΔS_irr for each source and for the universe. While I have calculated it for the universe I couldn't for the sources.

This is a simple example to help me to understand better the problem. I have other doubts which I'm going to specify below:

1) In equation (1) is T the temperature of the system or od the environment? (I think it's system temperature but I also read it refers to envirnoment)

2) Equation (1) can be also written as:

T*dS=dQ (2)

In this equation I think I'm considering an infinitesimal step of a trasformation where system are exchanging with a source at temperature T (wichi is the same temperature of system). If I consider a irreversible process equation (2) becomes:

T*dS=dQ+dS_irr(q)+dS_irr(L_f) (3)

dS_irr(q)=contribution of heat

dS_irr(L_f)= contribution of friction

However I really don't understand equation (3). First of all, if it's an irresversible transformation how can I consider an infinitesimal process? During an irreversible transformation I can't know the state of system because thermodinamics properties are not defined.

Also, since it's an irreverisble transformation and source and system could be different temperatures, what does T refers to?

Then, I have read that the term [dQ+dS_irr(q)] is equivalent to the heat of the reversible transformation.

Indeed I have read:

(dQ/T_source)+dS_irr(q)=(dQ/T)

where (dQ/T) refers to the equivalent reversible transformation and T should be the temperature of system.

I know that some passages could be not so clear but I did my best especially I don't speak English very well. I hope you can help me especially with equation (3) because I don't know its meaning at all.

thank you

I know that if A and B are two possible states of a system, the equation whcih defines variation of entropy from A to B is:

ΔS=∫(dQ/T)_rev (1)

So I have to find a generic reversible transformation, which go from A to B, and resolve the integral along this transformation. In this case system would evolve through infinite sources which are step by step at the same temperature of system.

I have thought to this example where two heat sources at different temperatures (T_h>T_c) exchange an amount of heat (Q).

As reversible transformations I have choosed isotermal transformations at the two different temperatures of the sources (you can see my solution in the pic below).

Now I start to have some problems. Let's consider the equation (1), that integral can be calculated also on the real (irreversible) transformation which system follows and we have that:

ΔS=∫(dQ/T)_rev≥∫(dQ/T)_irr (2)

In general it can be written:

∫(dQ/T)_rev=∫(dQ/T)_irr+ΔS_irr

where the term ΔS_irr is a contribution due to irreversible process.

What I have tried to do is to calculate ΔS_irr for each source and for the universe. While I have calculated it for the universe I couldn't for the sources.

This is a simple example to help me to understand better the problem. I have other doubts which I'm going to specify below:

1) In equation (1) is T the temperature of the system or od the environment? (I think it's system temperature but I also read it refers to envirnoment)

2) Equation (1) can be also written as:

T*dS=dQ (2)

In this equation I think I'm considering an infinitesimal step of a trasformation where system are exchanging with a source at temperature T (wichi is the same temperature of system). If I consider a irreversible process equation (2) becomes:

T*dS=dQ+dS_irr(q)+dS_irr(L_f) (3)

dS_irr(q)=contribution of heat

dS_irr(L_f)= contribution of friction

However I really don't understand equation (3). First of all, if it's an irresversible transformation how can I consider an infinitesimal process? During an irreversible transformation I can't know the state of system because thermodinamics properties are not defined.

Also, since it's an irreverisble transformation and source and system could be different temperatures, what does T refers to?

Then, I have read that the term [dQ+dS_irr(q)] is equivalent to the heat of the reversible transformation.

Indeed I have read:

(dQ/T_source)+dS_irr(q)=(dQ/T)

where (dQ/T) refers to the equivalent reversible transformation and T should be the temperature of system.

I know that some passages could be not so clear but I did my best especially I don't speak English very well. I hope you can help me especially with equation (3) because I don't know its meaning at all.

thank you