Thermodynamics Homework Questions

[mmmboh]

Homework Statement

Hi there, we got about 10 questions for an assignment and I have done most of them although there are a few I am unsure about.

1. Which of the following are extensive and which are intensive? The magnetic moment of gas, the electric field in a solid, the length of a wire, the surface tension of an oil film?

2. A mixture of hydrogen and oxygen is isolated and allowed to reach a state of constant temperature and pressure. The mixture is exploded with a spark of negligible energy and again allowed to come to a state of constant temperature and pressure. Is the initial state an equilibrium state? explain. Is the final state an equilibrium state? explain.

3. Describe how a system containing 2 gases can be in:
a)Mechanical but not thermal and chemical equilibrium
b)Thermal but not mechanical or chemical equilibrium
c)Thermal and mechanical equilibrium but not chemical

4. On a graph of volume versus temperature draw and label an isothermal expansion, an isothermal compression, an isochoric increase in temperature.

5. Is classical thermodynamic reasoning alone enough to determine
a) the average velocity of the molecules of a gas
b) the relation between the pressure dependence of the specific heat capacity of a solid and the temperature dependence of its volume.
c) the magnitude of the magnetic moment of a gas
d) the relation between the pressure and temperature of electromagnetic radiation in a cavity.
e) The magnitude of the specific heat capacity of a solid.
Briefly justify your answers.

The Attempt at a Solution

1. For the magnetic moment of a gas, I put extensive, for the electric field in a solid I put intensive, for the length of a wire I put intensive, and for the surface tension of an oil film I put extensive. Is this all correct?

2. Ok for this question I reasoned that a state of constant temperature and pressure means it is in thermal and mechanical equilibrium, satisfying two out of the three conditions. So in the initial state I thought it wasn't in chemical equilibrium because the hydrogen and oxygen atoms would be reacting but very slowly because they don't have enough energy to convert to water. (I'm not sure if that means they aren't in chemical equilibrium though, maybe that just means they aren't reacting at all and so are in equilibrium?). For the final state I reasoned that it was in equilibrium because it was in thermal and mechanical equilibrium, as well as chemical because the spark allowed the hydrogen and oxygen to convert to water.

Is this correct? I'm not even sure if it does convert to water because it says the spark has negligible energy, was if it wasn't negligible then would it not be in equilibrium? maybe the question has nothing to do with water and I am completely wrong I am not even sure.

3a) A system containing two gases can be in mechanical but not thermal or chemical equilibrium if the gases aren't moving, however the temperature at every point in the system isn't equal and the gases are not finished reacting with each other.
b) A system can be in thermal but not mech. of chem. equilibrium if the temperature at every point in the system is the same however the gases are moving and the are not finished reacting with each other.
c) A system can be in thermal and mech. equilibrium but not in chem. if the temperature is the same everywhere and the gases are not moving but they are still reacting.

4. For the isothermal expansion I drew temperature on the y-axis and volume on the x-axis, and drew a line of constant temperature in the increasing x-direction.
For the isothermal compression I drew the same thing as the isothermal expansion except I drew the line in the decreasing x direction starting from 0.
For the isochoric increase in temperature I drew the volume on the y-axis and the temperature on the x-axis and drew a line of constant volume in the increasing x-direction.

Is this right?

5. Ok for a) I am just not sure, does it have something to do with PV=nRT, and that pressure is F/A and that F is change in momentum over change in time? and so the answer would be yes? For b) Does this also have to do with PV=nRT? c)I would say no, I don't know what magnetism has to do with thermodynamics, help? d) I'm really not sure about this one, I would guess yes. e) I know C=Q/(mt)...so I would say yes, but I am not sure how to explain it.

Thanks for any help :)

 

[mmmboh]

Anyone?

 

[mathman44]

Hey again,

I'm having mainly the same problems as you are. for #1, I argued that the only extensive properties are the magnetic field and length.

for #5, I think the only one that is "no" is the first, thermodynamics is concerned only with macroscopic properties of matter. for "c", the magnetic moment of a gas is said to be "a property of thermodynamic systems" in the text, maybe that clarifies something for you? :s

#2/3 are really confusing. Any help?

 

[jegues]

Thermal equilibrium occurs when a system's macroscopic thermal observables have ceased to change with time.

Well for #2 the initial state given is definatly not a equilibrium state.

I'm not quite sure but I think that final state would be considered and equilibrium state, since a chemical reaction has already taken place I would assume it was to achieve a chemical equilibrium.

 

[Mapes]

The length of a wire isn't related to its mass?

 

[mmmboh]

Well it's not dependent on its mass, its mass is dependent on the length of the wire.

 

[Mapes]

So it's not clear why one would argue that length is an intensive quantity.

 

[mathman44]

Well for #2 the initial state given is definatly not a equilibrium state.

Why not? (Same question as #3.c)

 

[jegues]

Why not? (Same question as #3.c)

Thermal equilibrium occurs when a system's macroscopic thermal observables have ceased to change with time

Wouldn't a chemical reaction (explosion) be as a change? If it is then there is no way the initial state could have possibly been in equilibrium.

 

[mmmboh]

So it's not clear why one would argue that length is an intensive quantity.

Hmm I googled it, you appear to be right, apparently length is neither an intensive quality nor is it extensive...so you suggest i put neither?

 

[mathman44]

^^ I'm not sure... nothing would have changed if there was no spark. Why couldn't the mixture explode from equilibrium, assuming there is at least some pure hydrogen gas?

 

[jegues]

Why couldn't the mixture explode from equilibrium, assuming there is at least some pure hydrogen gas?

You raise a good point, as for the answer... I'm not entirely sure.

Perhaps someone else can clarify things for the both of us! :)

 

[Mapes]

Hmm I googled it, you appear to be right, apparently length is neither an intensive quality nor is it extensive...so you suggest i put neither?

It's either one or the other. If you combine two systems, is the total amount of the parameter equal, or doubled? This is how you tell if a parameter is intensive or extensive.

If you combine two systems that each have electric field E, the resulting electric field is E. The electric field is an intensive quantity.

If you combine two systems that each have surface tension $\gamma$ (energy per unit area), the resulting surface tension is $\gamma$. Surface tension is an intensive quantity.

If you attach two systems that have volume V, the resulting volume is 2V. Volume is an extensive quantity.

If you attach two systems that have length L...

 

[mmmboh]

Thanks, but for the length of the wire one, the question simply asks "is the length of a wire extensive or intensive", it doesn't give any other information, so how do you suggest I answer it?

I guess I should just put what you said about it...

 

[Mapes]

What do you think, based on my post #13?

 

[jegues]

If you combine two systems that each have electric field E, the resulting electric field is E. The electric field is an intensive quantity.

If you combine two systems that each have surface tension (energy per unit area), the resulting surface tension is . Surface tension is an intensive quantity.

If you attach two systems that have volume V, the resulting volume is 2V. Volume is an extensive quantity.

If you attach two systems that have length L...

This should be more than enough to answer your question

 

[mmmboh]

It is, thanks :)

 

[mmmboh]

Any help with the others?

 

[Mapes]

Which ones are still giving you problems?

 

[mathman44]

Questions 2 and 3 for me (post 12...)

 

[mmmboh]

Yeah those, and I'm not sure if my number 5 is right either.

 

[bp_psy]

If you combine two systems that each have electric field E, the resulting electric field is E. The electric field is an intensive quantity.

Could you explain this? Why wouldn't the Electric fields add?
I agree that E is intensive but I do not understand your reasoning in this case.

 

[Mapes]

Questions 2 and 3 for me (post 12...)

I agree with mmmboh's answers on 2 and 3. For question 5, it looks like you're both guessing about several of the answers. I'd recommend tracking the relationship or value that the question is asking about first, then checking whether it's derived from classical thermo.

 

[Mapes]

Could you explain this? Why wouldn't the Electric fields add?
I agree that E is intensive but I do not understand your reasoning in this case.

The electric field is generally described by a change in potential per unit length. Potentials combine linearly and so do lengths, so the electric field remains constant. Does this make sense?

 

[mathman44]

I don't understand why the system is considered out of equilibrium in it's initial state :S

 

[Mapes]

I don't understand why the system is considered out of equilibrium in it's initial state :S

It's just as mmmboh described; there was a driving force for reaction, which occurred slowly until a spark catalyzed it to completion. If the original system were already at chemical equilibrium, a small spark would have no effect. Plus, we already know that hydrogen and oxygen tend to react to form water.

 

[mathman44]

Ah; I wasn't aware that water was the equilibrium. Thanks.

 

[mmmboh]

Ok well the first one is no because thermodynamics is only concerned with macroscopic properties of matter, for the magnetude of the specific heat capacity of a solid I would say yes since "the change in the internal energy of a closed thermodynamic system is equal to the sum of the amount of heat energy supplied to or removed from the system and the work done on or by the system or we can say " In an isolated system the heat is constant"."...and specific heat capacity is the amount of heat required to raise a unit quantity of a substance one degree...I'm still not sure how to explain the magnetic dipole one, or the the relation between the pressure and temperature of electromagnetic radiation in a cavity one, or the relation between the pressure dependence of the specific heat capacity of a solid and the temperature dependence of its volume...I know PV=nRT, and if you hold pressure constant and raise the temperature then the volume will raise...I have tried looking I can't find much :S

 

[Mapes]

For 5b, think: Maxwell relations.

 

[mmmboh]

We actually haven`t done Maxwell relations yet...

 

[bp_psy]

Ok well the first one is no because thermodynamics is only concerned with macroscopic properties of matter, for the magnetude of the specific heat capacity of a solid I would say yes since "the change in the internal energy of a closed thermodynamic system is equal to the sum of the amount of heat energy supplied to or removed from the system and the work done on or by the system or we can say " In an isolated system the heat is constant"."...and specific heat capacity is the amount of heat required to raise a unit quantity of a substance one degree...I'm still not sure how to explain the magnetic dipole one, or the the relation between the pressure and temperature of electromagnetic radiation in a cavity one, or the relation between the pressure dependence of the specific heat capacity of a solid and the temperature dependence of its volume...I know PV=nRT, and if you hold pressure constant and raise the temperature then the volume will raise...I have tried looking I can't find much :S

In section "1.1 Scope of Thermodynamics" it is written that "From the principles of thermodynamics one can derive the general relations between such as coefficients of expansion,compressibilities,specific heat capacities ... The actual magnitude of quantities like those above can only be calculated on the basis of a molecular model."

 

[mmmboh]

In section "1.1 Scope of Thermodynamics" it is written that "From the principles of thermodynamics one can derive the general relations between such as coefficients of expansion,compressibilities,specific heat capacities ... The actual magnitude of quantities like those above can only be calculated on the basis of a molecular model."

So this means that the magnitude of the magnetic moment of a gas is also no, right? because that can only be calculated with statistical thermodynamics I think...

 

[mathman44]

I think you're right.

 

[mmmboh]

I hope so, it's too bad the book does such a bad job answering the questions and he doesn't cover anything on the assignments in class :S