Constant volume gas thermometer¶
Code: #121-000
File: apps/ideal_gas/gas_thermometer.ipynb
Run it online: 
The aim of this notebook is to show how the empirical temperature can be defined using a constant volume gas thermometer.
Interface¶
The main interface (main_block_121_000) is divided in two VBox: left_block_121_000 and center_block_121_000.
left_block_121_000 contains the following widgets: T_slider, gas_dropdown, p_pt_slider, Theta_text, save_button, clear_button and show_lines_check.
center_block_121_000 contains the bqplot figure fig_121_001.
[2]:
from IPython.display import Image
Image(filename='../../static/images/apps/ideal_gas/121-000.png')
[2]:
CSS¶
A custom css file is used to improve the interface of this application. It can be found here.
[ ]:
from IPython.display import HTML
display(HTML("<head><link rel='stylesheet' type='text/css' href='./../../static/custom.css'></head>"))
display(HTML("<style>.container { width:100% !important; }</style>"))
display(HTML("<style>.widget-label { display: contents !important; }</style>"))
display(HTML("<style>.slider-container { margin: 12px !important; }</style>"))
display(HTML("<style>.jupyter-widgets { overflow: auto !important; }</style>"))
Packages¶
[ ]:
import numpy as np
from bqplot import *
import bqplot as bq
import bqplot.marks as bqm
import bqplot.scales as bqs
import bqplot.axes as bqa
import ipywidgets as widgets
Physical functions¶
This are the functions that have a physical meaning:
get_vget_thetaget_theta_values
[ ]:
def get_v(p_pt, gas):
'''
This functions calculates the v value of a given gas
for a given p_pt and T_pt by solving the Van der Waals Equation.
T_pt is defined by convention as 273.16K.
$p_{pt} v^3 -(RT_{pt}+p_{pt}) v^2 +av -ab = 0 $
Inputs:
p_pt: float value for the pressure of the gas, when it is in thermal equilibrium with the Triple Point of Water
gas: integer index to characterize a gas
Returns:
v: float value for the volume of the gas corresponding to the given p_pt and T_pt
'''
a = a_data[gas]
b = b_data[gas]
poly = np.poly1d([p_pt, -R*T_pt -p_pt*b, a, -a*b])
solutions = np.roots(poly) #This statement uses numpy roots to solve third order polynomial equation
v = 0.0
for sol in solutions: #This control statement selects the only real solution.
if abs(sol.imag) < 1e-5:
v = sol.real
return v
[ ]:
def get_Theta(p_pt,T,gas):
'''
This function calculates the $\Theta$ (empirical temperature) of a given gas
for a given values of p and p_pt by applying the definition of empirical temperature.
$\Theta = \lim_{p_{pt} \rightarrow 0} 273.16K \frac{p}{p_{pt}}ç$
Inputs:
p_pt: float value for the pressure of the gas, when it is in thermal equilibrium with the Triple Point of Water
T: float value for the real temperature for which empirical temperature is being calculated
gas: integer index to characterize a gas
Returns:
Theta:float value for the empirical temperature corresponding to gas i and pressures p_pt and p
'''
a = a_data[gas]
b = b_data[gas]
v = get_v(p_pt,gas)
if abs(v - 0.0) < 1e-5:
Theta = T
else:
p = (R*T/(v-b)-a/v**2)
Theta = (p/p_pt * 273.16)
return Theta
[ ]:
def get_Theta_values(p_pt_values,T):
'''
This function applies iteratively the get_Theta function
to get a 2d numpy array of the Theta values for every gas and
p_pt. This array is used as a mark to generate the lines
Inputs:
p_pt_values: 1d numpy array for the p_pt values for which we are calculating Theta
T: float value for the real temperature for which empirical temperature is being calculated
Returns
Theta_values: 2d numpy array for the Theta values for each gas (first index) and each p_pt(second index)
'''
Theta_values = np.empty((3,n_of_points)) # This statement generates an empty 2d numpy array with correct dimensions.
for gas in [0,1,2]:
for i in range(n_of_points):
Theta_values[gas,i] = get_Theta(p_pt_values[i],T,gas)
return Theta_values
Main interface¶
[ ]:
# Fixed Parameters:
R = 8.31446 # Gas constant in L*kPa/K/mol
T_pt = 273.16 # Temperature of fixed point (triple point of water) in K.
# Plot parameters:
p_min = 0.0
p_max = 120.0
T_min = 100.0
T_max = 400.0
# Gas data:
labels = ["Hidrogenoa","Helioa","Nitrogenoa"]#, "Oxigenoa", "Neona", "Argona"]
latex = ["$H_2$", "$He$", "$N_2$"]#, "$O_2$", "$Ne$", "$Ar$"]
a_data = [24.7100, 3.4600, 137.00]#, 138.2, 21.3500, 135.500]
b_data = [0.02661, 0.0238, 0.0387]#, 0.03186, 0.01709, 0.03201]
colors = ['#0079c4','#f09205','#21c400']#,'#dd4e4f', '#ae8ccd', '#86564b']
opacities = [0.7, 0.7, 0.7]
# Initial state:
gas=0
p_pt = 100.0
T = 373.16
Theta = get_Theta(p_pt,T,gas)
# Lines parameters
n_of_points = 2 # Integer parameter to control how many points are to be calculated for lines
p_pt_values= np.linspace(0.1,120.0,n_of_points)
Theta_values = get_Theta_values(p_pt_values,T)
#######################################
#######CREATE THE FIGURES##############
#######################################
fig_121_001 = bq.Figure(title='Tenperatura enpirikoak gas eta presio ezberdinetarako',
marks=[],
axes=[],
padding_x=0.0,
animation_duration=0,
legend_location='top-left',
legend_style= {'fill': 'white', 'stroke': 'grey'},
background_style= {'fill': 'white', 'stroke': 'black'},
fig_margin=dict(top=70, bottom=60, left=80, right=30),
layout = widgets.Layout(width='100%'),
toolbar = True,
)
scale_x = bqs.LinearScale(min = p_min, max = p_max, allow_padding = False)
scale_y = bqs.LinearScale(min = T-0.2, max = T+0.2)
axis_x = bqa.Axis(scale=scale_x,
tick_format='.0f',#'0.2f',
tick_style={'font-size': '15px'},
tick_values = np.linspace(p_min, p_max, 7),
#num_ticks=7,
grid_lines = 'none',
grid_color = '#8e8e8e',
label='p_pt (kPa)',
label_location='middle',
label_style={'stroke': 'black', 'default-size': 35},
label_offset='50px')
axis_y = bqa.Axis(
scale=scale_y,
tick_format='.1f',#'0.2f',
tick_style={'font-size': '15px'},
num_ticks=5,
grid_lines = 'none',
grid_color = '#8e8e8e',
orientation='vertical',
label='Theta (K)',
label_location='middle',
label_style={'stroke': 'red', 'default_size': 35},
label_offset='50px')
fig_121_001.axes = [axis_x,axis_y]
#######################################
#########CREATE THE MARKS##############
#######################################
x_values = [ p_pt_values for i in range(3)]
y_values = Theta_values
current_point = bqm.Scatter(
x = [p_pt],
y = [Theta],
scales = {'x': scale_x, 'y': scale_y},
opacities = [1.0],
visible = True, #True, #t == '1.00',
colors = [colors[gas]],
labels = [labels[gas]],
display_legend = False,
tooltip = bq.Tooltip(fields=['x', 'y'], labels=['p_pt:', 'Theta:'], formats=['.2f', '.2f'])
)
saved_points_0 = bqm.Scatter(
x = [],
y = [],
scales = {'x': scale_x, 'y': scale_y},
opacities = [1.0],
visible = True, #True, #t == '1.00',
colors = [colors[0]],
labels = [labels[0]],
display_legend = True
)
saved_points_1 = bqm.Scatter(
x = [],
y = [],
scales = {'x': scale_x, 'y': scale_y},
opacities = [1.0],
visible = True, #True, #t == '1.00',
colors = [colors[1]],
labels = [labels[1]],
display_legend = True
)
saved_points_2 = bqm.Scatter(
x = [],
y = [],
scales = {'x': scale_x, 'y': scale_y},
opacities = [1.0],
visible = True, #True, #t == '1.00',
colors = [colors[2]],
labels = [labels[2]],
display_legend = True
)
lines = bqm.Lines(
x = x_values,
y = y_values,
scales = {'x': scale_x, 'y': scale_y},
opacities = opacities,
visible = False, #True, #t == '1.00',
colors = colors,
labels = labels,
display_legend = False
)
fig_121_001.marks = [current_point, saved_points_0, saved_points_1, saved_points_2, lines]
#######################################
#########CONTROL WIDGETS###############
#######################################
## Left Block (temperature, pressure and gas controls)
T_slider = widgets.FloatSlider(
min=100.0,
max=400.0,
step=1.0,
value=T,
description='T',
disabled=False,
continuous_update=True,
orientation='horizontal',
readout=True,
readout_format='.1f'
)
T_slider.observe(update_temperature,'value')
Gas_dropdown = widgets.Dropdown(
options=[('Hidrogenoa',0), ('Helioa',1), ('Nitrogenoa',2)],
value=0,
description='Gasa:',
disabled=False,
)
Gas_dropdown.observe(update_current_point, 'value')
p_pt_slider = widgets.FloatSlider(
min=p_min,
max=p_max,
step=0.1,
value=100.0,
description='p_pt',
disabled=False,
continuous_update=True,
orientation='horizontal',
readout=True,
readout_format='.1f'
)
p_pt_slider.observe(update_current_point,'value')
Theta_text = widgets.Label(
value = '%.2f' % Theta
)
save_button = widgets.Button(
description='Gorde puntua',
disabled=False,
button_style='', # 'success', 'info', 'warning', 'danger' or ''
tooltip='Click me'
)
save_button.on_click(save_point)
clear_button = widgets.Button(
description='Ezabatu puntuak',
disabled=False,
button_style='', # 'success', 'info', 'warning', 'danger' or ''
tooltip='Click me'
)
clear_button.on_click(clear)
show_lines_check = widgets.Checkbox(
description='Erakutsi lerroak',
disabled=False,
value=False,
)
show_lines_check.observe(show_lines, 'value')
#######################################
############ LAYOUT #################
#######################################
Temperature_Box = widgets.HBox([widgets.Label(value='$\Theta$'), Theta_text, widgets.Label(value='$K$')])
left_block_121_000 = widgets.VBox([], layout=widgets.Layout(width='30%', align_items='center'))
left_block_121_000.children = [widgets.HTML(value='<b>Neurtu nahi den temperatura (gas idealaren eskalan)</b>'),
T_slider, Gas_dropdown, p_pt_slider, Temperature_Box, save_button, clear_button, show_lines_check]
centre_block_121_000 = widgets.VBox([], layout=widgets.Layout(width='70%', align_items='center'))
centre_block_121_000.children = [fig_121_001]
main_block_121_000 = widgets.HBox([], layout=widgets.Layout(width='100%', align_self='center', align_items='center'))
main_block_121_000.children = [left_block_121_000, centre_block_121_000]
main_block_121_000