Correct statement about internal energy

[songoku]

Homework Statement

Which statement about internal energy is correct?
A. When two system have the same internal energy, they must be at the same temperature
B. When the internal energy of a system is increased, its temperature always rises
C. The internal energy of a system depends only on its temperature
D. The internal energy of a system can be increased without transfer of energy by heating

Homework Equations

U = PE + KE

ΔU = Q - W

The Attempt at a Solution

I think the answer is D my not sure about the reasoning. If let say I rub a metal against rough floor, the friction will increase the temperature of the metal hence the internal energy increase. Is this example of increasing internal energy without heating?

Thanks

 

[Delta2]

D is the correct answer according to me too but i had another example in my mind: Adiabatic compression of a gas, then we have $Q=0$ and $\Delta U=-W$, that is the internal energy increases due to the mechanical work $W$ from the environmental surroundings to the gas.

 

[haruspex]

Is this example of increasing internal energy without heating?

No, I think they are looking for a counterexample to the other options, i.e. an increase in internal energy without a temperature change.

 

[songoku]

No, I think they are looking for a counterexample to the other options, i.e. an increase in internal energy without a temperature change.

Example mentioned by @Delta2 suits option D or maybe not because for ideal gas change in internal energy corresponds to change in temperature?

 

[Delta2]

Example mentioned by @Delta2 suits option D or maybe not because for ideal gas change in internal energy corresponds to change in temperature?

I believe my example is correct regardless of how the internal energy of an ideal gas depends on temperature. Option D says that "can be increased" which means that if you find one example that satisfies it, it is enough.

@haruspex is just saying that maybe you should provide counter examples for the cases A to C, but i don't believe that is needed either.

Anyway one counterexample for B is for example we have a system that its internal energy increases due to flow of additional mass into the system at the same temperature. The temperature remains constant but because mass increases the internal energy also increases.
You can find similar counterexamples for A and C by considering the mass of the system...

 

[haruspex]

@haruspex is just saying that maybe you should provide counter examples for the cases A to C, but i don't believe that is needed either

I am saying that since D contradicts the other options an example of D is necessarily a counterexample of the others.

 

[rude man]

I go with D also. It's undoubetdly correct, per post #2. But I woukld have thought B also to be correct. It certainly is for an ideal gas where internal energy is solely an increasing function of temperature and I would have thought that fact extended more or less for other substances.

 

[haruspex]

I would have thought B also to be correct.

How about moving two magnets further apart? Wouldn't that increase internal energy, with no temperature change?

 

[Chestermiller]

I go with D also. It's undoubetdly correct, per post #2. But I woukld have thought B also to be correct. It certainly is for an ideal gas where internal energy is solely an increasing function of temperature and I would have thought that fact extended more or less for other substances.

Definitely not. It is also a function of specific volume. $$dU=C_VdT-\left[P-T\left(\frac{\partial P}{\partial T}\right)_V\right]dV$$

 

[rude man]

Definitely not. It is also a function of specific volume. $$dU=C_VdT-\left[P-T\left(\frac{\partial P}{\partial T}\right)_V\right]dV$$

If you solve for dT in the above is it really possible for U to increase without increasing T in a real system? Have you looked at this with the van der Waal equation for example? Or is it maybe true for liquids and/or solids exclusively?

 

[Chestermiller]

If you solve for dT in the above is it really possible for U to increase without increasing T in a real system? Have you looked at this with the van der Waal equation for example? Or is it maybe true for liquids and/or solids exclusively?

van der Waals equation: $$P=-\frac{a}{V^2}+\frac{RT}{(V-b)}$$
$$P-T\left(\frac{\partial P}{\partial T}\right)_V=-\frac{a}{V^2}$$

 

[Bilal Rajab Abbasi]

No D is probably not the answer..because friction "does" involve transfer of heat energy itself.
C: As temperature is the average Kinetic Energy of all the particles in the body.. your first equation denies this fact. As you are adding P.E to K.E to get U
B: Yes. When the temperature of a body is increase, internal energy does increase because as temperature increases the particles will vibrate and move with more Kinetic Energy hence they will collide with each other and the surrounding plus their own Energy increases.
A: Yes and No both...because Heat(Internal) is Kinetic Energy at the microscopic level. So, the increase in Kinetic Energy of one, let's say photon, will be negligible...
Please correct me If I am wrong..
Thanks!

 

[Chestermiller]

No D is probably not the answer..because friction "does" involve transfer of heat energy itself.

Who said anything about friction in D?

 

[Bilal Rajab Abbasi]

Who said anything about friction in D?

Um, see The explanation at the end please

 

[Chestermiller]

Um, see The explanation at the end please

As I showed in posts #9 and 11, the internal energy of a gas can increase even if its temperature remains constant. The internal energy is the sum of the kinetic energy of the molecules and their potential energy of mutual interaction. When a real gas is compressed sufficiently, the molecules come close enough together to increase the potential energy of mutual interactions, and therefore internal energy.

 

[haruspex]

Um, see The explanation at the end please

That was only songoku's guess at how D could work. That guess can be wrong without making D wrong.

 

[songoku]

I am really sorry for late reply.

How about moving two magnets further apart? Wouldn't that increase internal energy, with no temperature change?

There is internal energy between two magnets? Can you please elaborate more about it?

Thanks

 

[haruspex]

I am really sorry for late reply.
There is internal energy between two magnets? Can you please elaborate more about it?

Thanks

In a system consisting of two magnets, wouldn't increasing their PE constitute adding internal energy to the system!

 

[Chestermiller]

In a system consisting of two magnets, wouldn't increasing their PE constitute adding internal energy to the system!

In the more general version of the first law of thermodynamics, gravitational potential energy and macroscopic kinetic energy are treated separately from the internal energy U: $$\Delta E=\Delta U+\Delta (PE)+\Delta (KE)=Q-W$$ I think the same would apply to two magnets.

In the OPs first post, he mistakenly assumed that the U in the equation U=KE+PE is the same as the internal energy function U in the first law of thermodynamics. The U in the thermodynamic equation is the internal energy associated with small scale molecular interactions and random molecular kinetic energy.

 

[Apashanka]

D. The internal energy of a system can be increased without transfer of energy by heating

if the system is compressed isobarically and adibatically then the internal energy increases .
e.g ΔU=-pΔV,
where ΔV <0

 

[Apashanka]

B. When the internal energy of a system is increased, its temperature always rises

For ideal gas it is true as dU=C_vdt,
But may not be true for other gases.

 

[Apashanka]

A. When two system have the same internal energy, they must be at the same temperature

Same for ideal gas but may not be true for other gas.

 

[DrClaude]

For ideal gas it is true as dU=C_vdt,
But may not be true for other gases.

There is a very common situation where the internal energy of a system changes while the temperature stays constant. Thinking in terms of gases only is not the way to go.

Same for ideal gas but may not be true for other gas.

So if I have two ideal gases in different quantities at the same temperature, they have the same internal energy?

 

[Apashanka]

There is a very common situation where the internal energy of a system changes while the temperature stays constant. Thinking in terms of gases only is not the way to go.
So if I have two ideal gases in different quantities at the same temperature, they have the same internal energy?

No mass need to be taken into account and also the specific heat at constt. volume

 

[DrClaude]

No mass need to be taken into account and also the specific heat at constt. volume

Who said anything about specific heat? The statement is

A. When two system have the same internal energy, they must be at the same temperature

This is a generic statement. It is important for anyone wanting to understand thermodynamics that temperature by itself says nothing about the internal energy of a system. It is crucial to understand that energy is an extensive quantity while temperature is intensive.

 

[songoku]

In a system consisting of two magnets, wouldn't increasing their PE constitute adding internal energy to the system!

Oh I see.

In the more general version of the first law of thermodynamics, gravitational potential energy and macroscopic kinetic energy are treated separately from the internal energy U: $$\Delta E=\Delta U+\Delta (PE)+\Delta (KE)=Q-W$$ I think the same would apply to two magnets.

In the OPs first post, he mistakenly assumed that the U in the equation U=KE+PE is the same as the internal energy function U in the first law of thermodynamics. The U in the thermodynamic equation is the internal energy associated with small scale molecular interactions and random molecular kinetic energy.

Yeah I always think they are the same

I think I get it. Thank you very much for all the help