Equipment:
Eutectic is a mixture of two or more components that has unique properties different from those of the individual substances. Its melting point differs from the melting points of the individual substances, making it useful in metallurgy, pharmaceuticals, and even cooking.
In this problem, we will study the properties and obtain quantitative estimates of the observed effect.
In this part, we will consider heat exchange only with water to determine the effective thermal conductivity of the system.
To do this, you need to assemble the setup as shown in the photo below.
Instructions for data processing:
Data processing and storage will be done using MS Excel. You can immediately record all measured quantities in spreadsheets. For the jury to correctly check your work, name the files and record data in tables exactly as specified in the instructions.
On your computer's desktop, there is a folder named «J2». It contains the files «check.bat», «example.xlsx», and a folder named «First Name Last Name». You must rename the folder to include your last name and first name; otherwise, your work will not be checked!!!
After renaming the folder, open it. It contains subfolders with numbers corresponding to the sections they relate to, as well as a file «Report.docx». Do not change the names of the subfolders or the file «Report.docx»!
In the subfolder of each section, save all measurements and solution files.
The file «example.xlsx» is a template for recording measurements. All measured data must be formatted according to this template, named «MesX.xlsx», where X is the section number, and saved in the folder of that section. (For example, for question A1, the file with raw measurements should be named «MesA1.xlsx».) The measurement files should contain nothing but the raw measurements: alcohol concentration x, time measurements $t$, and temperature $T$.
All measurement files that are not formatted according to the template or contain extraneous data will not be evaluated!!!
Graphs with unlabeled axes WILL NOT be evaluated!
Solution files must contain comments explaining what is happening and being calculated in each column. These comments must also be written on the answer sheets for the corresponding section. Solution files must be named «SolX.xlsx», where X is the section number, and saved in the folder of that section. (For example, for question A2, the solution file should be named «SolA2.xlsx».)
See an example of a filled measurement file and a correctly formatted solution file below.
It is recommended to record the temperature at regular intervals to simplify further processing.
You must copy the file «example.xlsx» into the folder «A1» and rename it to «MesA1.xlsx». Fill the table according to the template in this file.
The derivative of temperature $T$ with respect to time is the quantity $dT/dt$, equal to the rate of change of temperature at a given moment. Due to the sufficiently large number of measurements, the value of $dT/dt$ at time $t$ can be calculated as the change in temperature over a small time interval $\Delta t$, divided by its duration, i.e.:
\[\frac{dT}{dt}(t) \approx \frac{T(t + \Delta t/2) - T(t - \Delta t/2)}{\Delta t}.\]As an example, using Excel cell operations, let's calculate the rate of temperature change $dT/dt$ at time $t=10~\mathrm{s}$(see figure).
In this example, columns A and B contain the values of time and temperature at those moments, respectively, and in column C, we want to record the values of the rate of temperature change $dT/dt$. For example, at time $t_{10} = 10 ~с$ , we consider the temperatures at times$t_0 = 0 ~с$ and $t_{20} = 20 ~с$, i.e.:
\[\frac{dT}{dt}(t_{10}) \approx \frac{T(t_{20}) - T(t_0)}{t_{20} - t_0} = \frac{29{,}2^\circ С - 20{,}1^\circ С}{20~с - 0~с} \approx 0{,}455^\circ С / с\](Instead of$0~\mathrm{s}$ and $20~\mathrm{s}$ values can be taken at $5~\mathrm{s}$ и $15~\mathrm{s}$ but this will increase the random error in calculating $dT/dt$).
The thermal power $P$, пtransferred from the water in the calorimeter to the contents of the syringe is governed by Newton's law of cooling:
\[
P = K \Delta T,
\]
where $K$ is the heat transfer coefficient.
Relate the rate of temperature change $dT/dt$of the water inside the syringe to the temperature of the water in the calorimeter $T_m$, of the water inside the syringe to the temperature of the water in the calorimeter $T$, $c_w$ - the specific heat capacity of water, $\rho_w$ - the density of water, and the volume $V$ of water in the syringe.
Assume that the temperature of the water in the calorimeter does not change during the measurements.
In this part, we will study the melting of a system consisting of two substances. Their solid mixture (i.e., alloy) forms a complex phase called a eutectic.
The phenomenon of eutectics is observed in any binary mixture. For example, the dependence of temperature on time during the cooling of a mixture of aluminum and copper looks like this:
Here on the graph:
Before the first kink (labeled $\alpha$-phase start), the entire mixture is in a liquid state.
Between the first and second kinks, solid copper particles and liquid eutectic coexist in the mixture.
After the second kink, the eutectic crystallizes, and the entire mixture becomes solid.
Note: In our problem, we study the point $T_m$ corresponding to the higher temperature, i.e., the moment when the entire contents become liquid.
The melting point of the eutectic is always lower than the melting points of both substances separately.
The phase diagram of the mixture usually looks like the picture below. The red arrow depicts the cooling process. The two kinks on the $T$ vs. $t$ graph correspond to phase transitions.
In this problem, we will study only the "upper" phase transition with temperature $T_m$.
First, assemble the setup as in the previous part, but this time you will need to place alcohol and acid inside the syringe.
To insert the thermocouple into the syringe, first heat it in hot water in the calorimeter to melt its contents.
We recommend placing the thermocouple near the middle of the syringe's height to eliminate edge effects and measure the desired temperature. Note that the melting points of the acid and alcohol, as well as any of their mixtures, lie in the range $40 - 80^\circ С$.
Remove the syringe from the hot water and let it cool in the air until the mixture solidifies. Once it has solidified, you can measure the rise in its temperature over time and thus study the phase transition.
For each series of measurements where the contents of the syringe change, use a new syringe and nozzle to avoid mixing solutions of different concentrations.
Qualitatively plot the dependence $T$ vs $t$ –temperature vs. time, as well as
$dT/dt$ vs $T$ – the rate of change of the temperature of the syringe contents vs. the value of this temperature.
It is recommended to set the temperature inside the calorimeter in the range $80 - 90 ^\circ С$. Try to observe heating over a sufficiently wide temperature range.
You must copy the file «example.xlsx» into the folder «B2» and rename it to «MesB2.xlsx». Fill the table according to the template.
Для проверки правильности сохранения исходных измерений вы можете запустить файл «check.bat». После исполнения этот файл должен выдать вам следующие строки в консоли:
If the file generates any errors or console errors, it means that not all files meet the formatting requirements and need to be corrected. If the file does not generate a series of measurements, it is also incorrectly recorded and saved!
Try to evenly cover the entire concentration range $x\in[0;1]$. It is recommended to set the temperature inside the calorimeter in the range $80^\circ-90^\circ\mathrm{C}$.
You must copy the file «example.xlsx» into the folder «B5» and rename it to «MesB5.xlsx». Fill the table according to the template.
Plot the dependence $T_m(x)$— melting temperature vs. concentration. Insert this graph into «Report.docx».
From the graph, estimate the minimum melting temperature of the eutectic $T_e$ of alcohol and acid and the concentration $x_e$ at which it is achieved. Record the values of $T_e$ and $x_e$ in the corresponding places in «Report.docx».
To check the correctness of saving the raw measurements, you can run the file «check.bat». After execution, this file should output the following lines in the console: