# Why entropy is an extensive quantity ?

1. Dec 19, 2009

### abcdefg10645

Why entropy is an extensive quantity ?

2. Dec 19, 2009

### sweet springs

Hi.
Entropy increase by volume. Gas of 2 m^3 have double entropy than gas of 1 m^3 in the same temperature and pressure.
Regards.

3. Dec 20, 2009

### abcdefg10645

But how do you know that "Entropy increase by volume. "

Let me restate my question :

How can ew know that entropy is an extensive quantity by its definition

dS=$$\int(dQ/T)$$

4. Dec 20, 2009

### sweet springs

Hi.
First law of thermodynamics, about the conservation of energy: δQ=dU - dW =dU - pdV
δQ is extensive because dU and pdV are extenxive. δQ/T and ∫δQ/T are also extensive.
Regards.

5. Dec 20, 2009

### abcdefg10645

But for different systems , their temperature T may not be the same !

First Law sates that deltaQ=dU+deltaW

We can only infer that deltaQ is additive !

Thanks

6. Dec 20, 2009

### sweet springs

Hi.

Thanks for correcting my sign mistake of deltaW.
You are right. Additive is the essence of extensive quantity. For isolated two different systems
S1 = ∫dQ1/T1, S2 = ∫dQ2/T2 and the entropy of total system is S = S1 + S2.
Regards.

7. Dec 20, 2009

### abcdefg10645

Ha ha~never mind ~I believe that were just some typos

But , I cannot understand why entropy is additive ?

8. Dec 20, 2009

### sweet springs

Hi.
Definition ∫　δQ/T is already addition of　δQ/T , isn't it ?
Regards.

9. Dec 20, 2009

### abcdefg10645

Using the integral to calculate the (delta)S "OF A SYSTEM" is to calculate the difference of entropy for a particular process , I want to ask why , for different systems , it can also be added ?

10. Dec 21, 2009

### sweet springs

Hi. How's that?

1 m^3 of liquid water of the room temperature and in standard pressure has entropy say S.
Let us remove half of water. Do you agree that the left water has entropy S/2 ?

We will bring removed water into another isolated room of the same temperature and pressure.
Do you agree that brought water has entropy S/2?

Water in the first room and water in the second room are different systems.
Do you agree that the integrated system of the two has entropy S?

In room 1 we increase the temperature then the entropy of water becomes S'/2
Do you agree that the integrated system of the two has entropy S/2 + S'/2 ?
Change of entropy ∫δQ/T for room 1 system is regarded as change of entropy in the integrated system, isn't it?