Thermodynamics problem

[vovka]

Hello!

The problem is stated as follows:

The heat capacity of a body, in the considered temperature interval, depends
on temperature according to the expression 

C=10+0.002T+3*10^{-5}T^{2}JK^{-1}

How much heat is released when the temperature varies from  T₁= 400 K to T₂=300 K ?

My way of solving:

we have  C=δQ/dT    so  δQ=CdT  . By integrating both sides on the given interval we should get the desired quantity.

My computation yields -1440J but the answer is -2.1kJ?! What's wrong?

One remark is that  δQ is not a differential so taking integral of it may be not correct.

Will be grateful for reasonable opinions.

Thanks for attention!

 

[studiot]

Can you check your question? 

In particular is the equation of state for the heat capacity in terms of temperature in kilojoules per degree K?

The equation of state can't be correct as it stands since the RHS has units of (degrees)² as stated.

It may be an English language problem, but there is something wrong with the information as written.

If it's any consolation you have your integration arithmetic correct, but that is not the issue.

 

Edited May 20, 2018 by studiot

[vovka]

Thanks for reply, studiot. The equation is as written in the textbook, \[{C=10+0.002T+3*10^{-5}T^{2} JK^{-1}}\] I think T plays role here as some independent variable in [300,400] and we should treat it as simple number but your argument is logical. So far I have discovered some misprints in this textbook thus it may be here as well. 

 

Edited May 21, 2018 by vovka

[studiot]

One of the difficulties is do they mean C_p or C_v?

I note the capital C which means the heat capacity for the whole body, not per kilgram, which would be denoted by a small c.

Your thermodynamic analysis is not correct, allthough it leads to the correct equation.

The State variable you require is the enthalpy, H.

[math]\Delta _{400}^{300}H = \int_{T = 400}^{T = 300} {{C_p}} dT[/math]

 

This integration does indeed give you q, the heat transferred by the first law since it does not include any work done.

(Can you show this?)

 

Using Cv on the otherhand would also include the work done as it refers to change of internal  energy (U or E).

 

In this integration the heat capacity is a function of temperature , so cannot be taken outside the integration as a constant.

 

 

 

[studiot]

It should be noted that the general heat capacity, C, is path dependant so your question nedds to supply more information.

[vovka]

Many thanks for your careful review, studiot. I agree we can express heat through  enthalpy \({δQ=dU+pdV=dH}\), and since the body is hardly compressible then variations in V and p are negligible so there is no work done. From (2.28) we have \({ dQ=dU=dH=CdT}\) that leads us to integration. We don't have to specify what kind of C we use since \({C≈C_V≈C_P}\).

Finally from the above I conclude that the solution approach is mainly correct and the discrepancy should be referred to misprinting or lack of info in the problem.

(taking \({C_V}\) implies V=const so pdV=0 and we shouldn't include any work done)

p.s. could you name the book you've posted above  

Edited May 22, 2018 by vovka

[studiot]

On 22/05/2018 at 11:02 AM, vovka said:

Many thanks for your careful review, studiot. I agree we can express heat through  enthalpy δQ=dU+pdV=dH , and since the body is hardly compressible then variations in V and p are negligible so there is no work done. From (2.28) we have dQ=dU=dH=CdT that leads us to integration. We don't have to specify what kind of C we use since C≈CV≈CP .

Finally from the above I conclude that the solution approach is mainly correct and the discrepancy should be referred to misprinting or lack of info in the problem.

(taking CV implies V=const so pdV=0 and we shouldn't include any work done)

p.s. could you name the book you've posted above  

 

Hi vovka, what are you studying?

 

The problem with internal energy is that substances expand on heating so work is done, so using Cv leads to false values.

 

That is why the old term for enthalpy was 'heat content'.

As regards the polynomial expansion of the variation of heat capacity with temperature, I noted the dimensional analysis problem.
This simply means that the constants a, b,c d, etc must have suitable physical MLT^oK dimensions to bring the equation to dimensional balance.

There are various versions of polynomials in use.

 

 

 

As regards the extracts I have

Heatcap 1 came from

Chemical Thermodynamics
Frederic T Wall  2nd ed 1965
Freeman San Francisco and London

This is a very good book, with lots of background explanation , but I think you might have trouble finding a copy today.

Heatcap 3 came from

Materials Thermodynamics
Chang and Oates
Wiley 2010

Heatcap 4 came from

Thermodynamics
Cengel and Boles
McGraw-Hill 1989

Heatcap 2 came from

Chemical Tehmodynamics
E F Caldin
Oxford University Press 1958

There are more useful pages to go with this, but I doubt you will find the older books, so if you let me have a PM with an email address that can receive jpegs I can let you have a few more scans of surrounding pages, at better quality if you want to print them out.

[vovka]

Hi! Really appreciate your wide thoughtful participation in this topic studiot. I study general physics ( thermodynamics  in particular). Thank you for kind proposition about scans but I was looking for textbook in thermal physics (thermodynamics and kinetic theory of gases). I have a few (though truly demanding but think worth to give a try) and the above mentioned Cengel and Boles seems to be good additional reference.

 

[studiot]

32 minutes ago, vovka said:

Hi! Really appreciate your wide thoughtful participation in this topic studiot. I study general physics ( thermodynamics  in particular). Thank you for kind proposition about scans but I was looking for textbook in thermal physics (thermodynamics and kinetic theory of gases). I have a few (though truly demanding but think worth to give a try) and the above mentioned Cengel and Boles seems to be good additional reference.

 

C & B is an engineering book (Mechanical). Other branches of Engineering have their own.

 

But for Physics I would look at the following list.

 

Basic Termodynamics

Gerald Carrington

Oxford University Press

 

An excellent thoroughly modern book about Classical Termodynamics including Gibbsean and Caratheodory formulations.
He does not, however treat Statistical Mechanics.

But all subjects treated take the reader from basics all the way through undergraduate and just into post grad level.

Statistical Thermodynamics

Andrew Maczek

Oxford University Press

Supplies the missing SM material

Both owe much to this book

 

Thermodynamics and Statistical Mechanics

A H Wilson

Cambridge University Press

"This account written primarily for theoretical physicists and for experimental physicists wwishing to enter more deeply into the fundamental principles of the subject."

 

Elements of Classical Thermodynamics for advanced students of Physics.

A B Pippard

Cambridge University Press

This small book offers a great deal of insight behinfd the scenes.

 

Finally in some European countries (France in particular) the subject is studied in a different way, as part of a wider 'materials' based subject.

The classic text here is by J Lemaitre (University of Paris) and Chaboche (Office National d'Etudes and des REcherches)

My English language translation was published by

Cambridge University Press

as

Mechanics of Solid Materials