Determination of Molar Mass from freezing point¶
The aim of the experiment if to calculate Molar Mass of compound (urine) after determination of its freezing point ($T_{fr}$) and to determine Vant-Hoff constant ($i$) for compounds under study. The $T_{fr}$ is measured with kryoscopy method, when solution is being cooled down with thermostate, and solution's tempreature is being measured with thermocouple. When $T_{fr}$ is achieved, usually over-freezing is observed in the time-temperature curve. The plateau after over-freeze corresponds to real freezing temperature of the solution.
Throughout the experiment several referenced solutions are measured: distilled water and KCl. The $T_{fr} (water)$ is taken as calibration point for equipment, whereas KCl serves as checkpoint for the experiment, since theoretically $i(KCl)$ is approximately 2. Finally, solutions of urine with two different concentrations are measured. Given the high dilution of solutions, the equations suitable for ideal solutions are applicable as it will be shown further. It allows to say that $T_{fr}$ changes linearly with concentration ($m$).
Import libraries¶
import numpy as np # math
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit # fitting custom function
from matplotlib.dates import datestr2num
from tabulate import tabulate
Get seconds from time¶
def get_sec(time_str):
h, m, s = time_str.split(':')
return int(h) * 3600 + int(m) * 60 + int(s)
Find the plateau (index) from the raw data¶
The plateau is determined as line, where temperature doesn't change more than by 0.5
def find_plateau_index(y, threshold):
for i in reversed(range(len(y))):
if y[i] - threshold <= y[i-1] <= y[i] + threshold:
last_index = i
else:
break
return last_index
Define compounds¶
compounds = {}
compounds["water_stir"] = { "filename": "2405_water_stir.txt",
"title": "Water stir",
"plot_fmt": "r",
"plot_pos_x": 0,
"plot_pos_y": 0,
"calculate_molar_mass": False }
compounds["water_no_stir"] = { "filename": "2405_water_nostir.txt",
"title": "Water no stir",
"plot_fmt": "r",
"plot_pos_x": 0,
"plot_pos_y": 1,
"calculate_molar_mass": False }
compounds["kcl"] = { "filename": "2405_kcl.txt",
"title": "KCl",
"plot_fmt": "g",
"plot_pos_x": 1,
"plot_pos_y": 0,
"mass_solid": 1.0010,
"mass_water": 50.0023,
"calculate_molar_mass": True }
compounds["urea1"] = { "filename": "2405_urea1.txt",
"title": "Urea Concentrated",
"plot_fmt": "b",
"plot_pos_x": 1,
"plot_pos_y": 1,
"mass_solid": 1.0011,
"mass_water": 50.0078,
"calculate_molar_mass": True }
compounds["urea2"] = { "filename": "2405_urea05.txt",
"title": "Urea Diluted",
"plot_fmt": "y",
"plot_pos_x": 2,
"plot_pos_y": 0,
"mass_solid": 0.4999,
"mass_water": 50.0039,
"calculate_molar_mass": True }
Read files, extract time and temperature values, find plateau¶
for type, info in compounds.items():
time_abs, temperature = np.loadtxt(info["filename"], unpack=True, delimiter='\t', encoding = 'ascii', converters={0: get_sec})
time_rel = time_abs - time_abs[0]
idx = find_plateau_index(temperature, 0.05)
compounds[type]["temperature"] = temperature
compounds[type]["time_rel"] = time_rel
compounds[type]["temperature_plateau"] = temperature[idx-1:]
compounds[type]["time_rel_plateau"] = time_rel[idx-1:]
compounds[type]["freezing_temperature"] = np.round_(np.mean(compounds[type]["temperature_plateau"]), decimals = 2, out = None)
temperature_water = (compounds["water_stir"]["freezing_temperature"] + compounds["water_no_stir"]["freezing_temperature"])/2
Constants from the literature¶
Molar_mass_kcl = 74.55
Molar_mass_urea = 60.06
Kf_water = 1.855
Molar masses can be googled or easily calculated with help of periodic table. $K_f(water)$ is kryoscopic water constant, also known from temperature.
Molar mass calculation¶
def find_molar_mass(temperature_water, temperature_solution, Kf, mass_water, mass_solid):
Molar_mass = mass_solid * Kf * 1000 / ((temperature_water-temperature_solution) * mass_water)
return Molar_mass
for type, info in compounds.items():
if not info["calculate_molar_mass"]:
continue
temperature_solution = compounds[type]["freezing_temperature"]
mass_solid = compounds[type]["mass_solid"]
mass_water = compounds[type]["mass_water"]
molar_mass = find_molar_mass(temperature_water, temperature_solution, Kf_water, mass_water, mass_solid)
compounds[type]["mass"] = molar_mass
Molar mass is determined by the equation
$M = \frac{mass_{compound} \cdot K_f(water) \cdot 1000 }{ \Delta T_{fr} \cdot mass_{water}}$
where $\Delta T_{fr}=T_{fr}(water)-T_{fr}(solution)$
Determination of Vant-Hoff constant¶
mass_urea_avg = (compounds["urea1"]["mass"] + compounds["urea2"]["mass"])/2
mass_kcl_avg = compounds["kcl"]["mass"]
vanthoff_kcl = Molar_mass_kcl / mass_kcl_avg
vanthoff_urea = Molar_mass_urea /mass_urea_avg
In order to determine Vant-Hoff constant, we need to divide expected molar mass to experimentally obtained molar mass.
Resulting table¶
table = [['Name', 'Molar mass exp', 'Molar mass lit', 'Vant Hoff'],
['Urea', mass_urea_avg, Molar_mass_urea, vanthoff_urea],
['KCl', compounds["kcl"]["mass"], Molar_mass_kcl, vanthoff_kcl]]
print(tabulate(table, headers='firstrow', tablefmt='fancy_grid'))
Plot figure¶
figure_title = 'The freezing curves for compounds under study'
fig, axs = plt.subplots(3, 2)
for type, info in compounds.items():
data = compounds[type]
ax = axs[info["plot_pos_x"], info["plot_pos_y"]]
ax.plot(data["time_rel"], data["temperature"], data["time_rel_plateau"], data["temperature_plateau"], info["plot_fmt"])
ax.set_title(f'{info["title"]} $T_$ = {data["freezing_temperature"]}')
ax.set(xlabel='Time, sec', ylabel='Temperature, C')
fig.suptitle(figure_title, fontsize=16)
fig.canvas.manager.set_window_title(figure_title)
fig.delaxes(axs[2][1])
fig.tight_layout()
plt.show()
