How Much Liquid Nitrogen is Needed to Freeze Water at 0ºC?

[Ethan_Tab]

Homework Statement

*All units given are in standard SI units unless otherwise specified.*
*Assume a closed system*

How much of -200^ºC Liquid Nitrogen is required to completely freeze 200mL (0.2kg) of pure liquid water at 0.00^ºC (No change in heat, only a change in state)?

Given Values.
n= Liquid nitrogen
w= Liquid water
C= Specific heat capacity
L_f= Latent Heat of fusion
L_v= Laten Heat of vaporization

C_w=4.2E3
C_n= 1.1E2
L_vn=2.0E5
L_fw=3.3E5
Melting point of Nitrogen= -209.9^ºC
Condensation point of Nitrogen= -196.8^ºC

Homework Equations

Q=mcΔT
Q=mL<sub>v
Q=mLf
Conservation of thermal energy.

3. The Attempt at a Solution
Here is my thought process, Ideally, a logical and arithmetical check would be much appreciated.

Since we know the water cannot change temperature and instead, only states (L-S), we can state a "restriction" saying that all energy which will convert the unknown mass of nitrogen "mn" will be the absolute value of the energy the water will give off during the mw*Lfw process.

By extension, this also means that the all the stage changes which occur to the Nitrogen to get it to zero need to add up to mw*Lfw

So... (skipping two steps and common factoring the unknown mass N and isolating for it) we get:

Mn=[Mw*Lfw]/[Cn*(3.2)+Lvn+Cn*(196.8)]

Plugging in given values in section 1 I get mass as being 0.29729kg. If anyone could confirm this both numerically and the process itself logically, It would be truly appreciated.</sub>

 

[haruspex]

The specific heat for liquid nitrogen is unlikely to be the same as for gaseous nitrogen.
I'm a bit worried by the "closed system" condition. The vaporisation temperature (and, I would have thought, the latent heat) will depend on the ambient pressure, but in a closed system that will increase as the heat transfer proceeds. Maybe it is mainly there to indicate what specific heat to use for the gaseous nitrogen.

 

[Chestermiller]

The specific heat for liquid nitrogen is unlikely to be the same as for gaseous nitrogen.
I'm a bit worried by the "closed system" condition. The vaporisation temperature (and, I would have thought, the latent heat) will depend on the ambient pressure, but in a closed system that will increase as the heat transfer proceeds. Maybe it is mainly there to indicate what specific heat to use for the gaseous nitrogen.

The implication is that the process is carried out at atmospheric pressure. Otherwise, the given condensation temperature could be some other value. It is possible to do this process at 1 atm in a closed system (but not an isolated system) by allowing the volume to increase.

Chet

 

[Ethan_Tab]

The specific heat for liquid nitrogen is unlikely to be the same as for gaseous nitrogen.
I'm a bit worried by the "closed system" condition. The vaporisation temperature (and, I would have thought, the latent heat) will depend on the ambient pressure, but in a closed system that will increase as the heat transfer proceeds. Maybe it is mainly there to indicate what specific heat to use for the gaseous nitrogen.

The implication of the "closed system", is not literal. It is simply there to show that no heat is lost to the environment. As for the heat capacity of Nitrogen Vapour; would it not remain the same as that of liquid nitrogen? I remember learning that heat capacities remain the same for all substances other then water.

 

[Chestermiller]

The implication of the "closed system", is not literal. It is simply there to show that no heat is lost to the environment. As for the heat capacity of Nitrogen Vapour; would it not remain the same as that of liquid nitrogen? I remember learning that heat capacities remain the same for all substances other then water.

This is not correct. The term "closed system" has a precise meaning, requiring that no mass enters or leaves the system. So the term closed system is literal. Also, the heat capacity of nitrogen vapor is not the same as liquid nitrogen. What you learned about heat capacities remaining the same for all substances other than water is obviously incorrect. If you are not sure what you are talking about, please don't give misinformation on Physics Forums.

Chet

 

[Ethan_Tab]

This is not correct. The term "closed system" has a precise meaning, requiring that no mass enters or leaves the system. So the term closed system is literal. Also, the heat capacity of nitrogen vapor is not the same as liquid nitrogen. What you learned about heat capacities remaining the same for all substances other than water is obviously incorrect. If you are not sure what you are talking about, please don't give misinformation on Physics Forums.

Chet

Let me clarify what I meant to say: The physics course which I'm currently taking does not deal with pressures and as such the term "closed system", simply implies a system in which no heat is lost to the surroundings. With that being said, If internal pressures were negligible and the capacity of liquid nitrogen (which was specially given in the question) applied to all its states, would my work be considered correct?

 

[Chestermiller]

Let me clarify what I meant to say: The physics course which I'm currently taking does not deal with pressures and as such the term "closed system", simply implies a system in which no heat is lost to the surroundings. With that being said, If internal pressures were negligible and the capacity of liquid nitrogen (which was specially given in the question) applied to all its states, would my work be considered correct?

Sure, but it is a known fact that the heat capacity of liquid nitrogen is not the same as the heat capacity of nitrogen gas/vapor. Look it up on Google.

Chet

 

[haruspex]

The implication is that the process is carried out at atmospheric pressure. Otherwise, the given condensation temperature could be some other value. It is possible to do this process at 1 atm in a closed system (but not an isolated system) by allowing the volume to increase.

Chet

Ah yes, of course. It has to be interpreted as constant pressure, or there's not enough information.

 

[mfb]

In terms of experiments: a closed system is not a practical thing for an actual experiment, but a well isolated box with a small hole would give a very good approximation - nitrogen can leave it at 0°C where it would not participate in further heat exchange anyway.
=> yes, assume a closed system at constant pressure

 

[Chestermiller]

In terms of experiments: a closed system is not a practical thing for an actual experiment, but a well isolated box with a small hole would give a very good approximation - nitrogen can leave it at 0°C where it would not participate in further heat exchange anyway.
=> yes, assume a closed system at constant pressure

Conceptually, one can have an insulated massless frictionless piston, with air outside the insulated cylinder to provide the constant pressure.

Chet

 

[mfb]

That's exactly what I meant with "not a practical thing" :D.

 

[Chestermiller]

That's exactly what I meant with "not a practical thing" :D.

I agree. But, given the level of conceptual idealization inherent in the problem statement (after all, how often do we ever dump water ice into liquid nitrogen in real life?), this additional level of idealization is not much of a stretch, and is typical of the idealizations that are often invoked in thermo homework problems.

Chet

 

[haruspex]

Ah yes, of course. It has to be interpreted as constant pressure, or there's not enough information.

Now I'm having doubts. If you use C_p, doesn't that mean the gas is doing work as it expands? Where is that work going, if it's a closed system?

 

[Chestermiller]

Now I'm having doubts. If you use C_p, doesn't that mean the gas is doing work as it expands? Where is that work going, if it's a closed system?

In the terminology as it was taught to me (and most others), a closed system is one for which the only constraint is that no mass enters or leaves the system. A closed system is fully capable of exchanging heat and work with its surroundings. We were also taught that an isolated system is one that, in addition to not exchanging mass with its surroundings, also does not exchange heat or work. I realize that, in some books and disciplines, the term closed system is used to describe what I call an isolated system. So it is important to recognize that there is ambiguity in the terminology.

http://chemistry.about.com/od/chemistryglossary/g/Closed-System-Definition.htm

In the context of this problem statement, either knowingly or unknowingly, it appears that author implied the use of "closed system" in the manner that I learned it. So, under that definition, the process takes place at constant pressure, the expanding gas does work on the surroundings, and the change in enthalpy for the combined contents of the container is zero (since the vessel is insulated with no Q being exchanged).

Chet