Equilibrium temperature of some ice and steam

AI Thread Summary
The discussion revolves around calculating the equilibrium temperature of a system containing ice and steam. The initial calculations yielded incorrect results, prompting participants to explore potential errors in computational steps and the use of latent heat values. A key point raised is the sensitivity of the final temperature to the specific latent heat of vaporization used, with discrepancies noted between various sources. Participants suggest using Celsius for easier calculations and discuss the possibility of significant digit preferences affecting the results. The conversation highlights the importance of accurate data and careful handling of phase change calculations in thermodynamics.
ElPimiento
Messages
16
Reaction score
0

Homework Statement


"A well-insulated bucket of negligible heat capacity contains 120 g of ice at 0°C. If 20 g of steam at 100°C is injected into the bucket, what is the final equilibrium temperature of the system?"

Homework Equations


$$Q_{fus} = m_{water}L_{fus}$$
$$Q_{vap} = m_{steam}L_{vap}$$
$$Q = mc_{water}\Delta T$$
Where ##L_{fus} = 334 \frac{kJ}{kg}##, ##L_{vap} = -2230 \frac{kJ}{kg}## (since the phase change is from gas to liquid), and ##c_{water} = 4.184 \frac{kJ}{kgK}##

The Attempt at a Solution


(I'm beginning to suspect the error is a computational one, or that I've given some term the wrong sign)

Since the container is well-insulated I can write that no heat is lost:

$$\begin{align*}
\sum Q & = 0 \\
Q_{fus} + m_1c_{water}\big(T_f - (273K)\big) + Q_{vap} + m_2c_{water}\big(T_f - (373K)\big) & = 0 \\
\end{align*}$$

Where ##m_1## refers to the ice and ##m_2## refers to the steam. Some algebra,

$$\begin{align*}
\big(m_1c_{water}T_f - m_1c_{water}(273K)\big) & \\
+ \big(m_2c_{water}T_f - m_2c_{water}(373K)\big) & = -(Q_{fus} + Q_{vap}) \\ \\
\big(m_1c_{water}T_f + m_2c_{water}T_f\big) & \\
- \big(m_1c_{water}(273K) + m_2c_{water}(373K)\big) & = -(Q_{fus} + Q_{vap}) \\ \\
T_fc_{water}(m_1 + m_2) & = -(Q_{fus} + Q_{vap}) \\
&\ \ \ \ + \big(m_1c_{water}(273K) + m_2c_{water}(373K)\big) \\ \\
T_f & = \frac {-(Q_{fus} + Q_{vap}) + \big(m_1c_{water}(273K) + m_2c_{water}(373K)\big)} {c_{water}(m_1 + m_2)}.
\end{align*}$$

Numerically, this becomes,

$$\begin{align*}
T_f & \approx \frac {-(-4.52 kJ) + \big((137 kJ) + (31.2 kJ)\big)} {0.586 \frac{kJ}{K}} \\
& \approx 294 K.
\end{align*}$$

or 21.7°C; but this is incorrect.

Thanks,
Andrew
 
Physics news on Phys.org
ElPimiento said:
21.7°C; but this is incorrect.
I make it nearer 22. It looks like you used 273K converting to K but 272.7 converting back. Or maybe you accumulated some rounding errors.
There's no need to work in K here, easier in C:
((4.52/4.184 + 100 * 0.020)/(0.120+0.020).
 
I'm submitting the question online so here are the values I know are incorrect: 21.7°C, 21.8°C, 22°C, and 8.998°C.
So, I hope it is not the case that the margin of error is crazy small, I also hope there is nothing fishy going on with the answer to the question that the website is using...

Thanks for the suggestion about using C.
 
ElPimiento said:
I'm submitting the question online so here are the values I know are incorrect: 21.7°C, 21.8°C, 22°C, and 8.998°C.
So, I hope it is not the case that the margin of error is crazy small, I also hope there is nothing fishy going on with the answer to the question that the website is using...

Thanks for the suggestion about using C.
I got 22.00, so either the app is wrong or it is being very picky about significant digits. Maybe it wants 22.0, but I cannot see why that would be preferred to 22. The masses are not given as 20.0 etc.
 
haruspex said:
I got 22.00, so either the app is wrong or it is being very picky about significant digits. Maybe it wants 22.0, but I cannot see why that would be preferred to 22. The masses are not given as 20.0 etc.
I'm going to bring it to my professor's attention. Thanks for the second opinion!
 
I got 23.15 C but perhaps I made an error somewhere.
 
CWatters said:
I got 23.15 C but perhaps I made an error somewhere.
No, I think you may be right. I suspect a sign error in the handling the f the latent heats. Will check.
 
haruspex said:
No, I think you may be right. I suspect a sign error in the handling the f the latent heats. Will check.
Edit: no, I still get 22:
Heat left over from conversion of all to water = 20x2230-120x334=4520J.
(4520/4.184+100*120)/(120+20)=22.00218..
 
Think I might have found the problem. The calculation appears sensitive to the exact value used for the latent heat of vaporization.

If I use the figure from here..
https://en.wikipedia.org/wiki/Latent_heat
..which has the latent heat of vaporization as 2264.76 kJ/kg then I get a figure of around 23C.
 
  • #10
CWatters said:
Think I might have found the problem. The calculation appears sensitive to the exact value used for the latent heat of vaporization.

If I use the figure from here..
https://en.wikipedia.org/wiki/Latent_heat
..which has the latent heat of vaporization as 2264.76 kJ/kg then I get a figure of around 23C.
Ah.. I didn't check those data. Online I see 2230, 2257, 2260, 2.3 x 103. 2230 is rather an outlier.
 
Back
Top