Stability condition on van der Waals isotherms¶
Code: #11A-000
File: apps/van_der_waals/stability.ipynb
Run it online: 
The aim of this notebook is to visualize the \(\left(\frac{\partial p}{\partial v}\right)_{T,N} < 0\) stability condition on van der Waals isotherms.
Interface¶
The main interface (main_block_11A_000) is divided in two HBox: top_block_11A_000 and bottom_block_11A_000. bottom_block_11A_000 contains of 2 bqplot Figures: fig_11A_001 and fig_11A_002.
[1]:
from IPython.display import Image
Image(filename='../../static/images/apps/11A-000_1.png')
[1]:
The slider T_slider updates the values of \(T\) which updates the lines of fig_11A_001 and fig_11A_002.
[2]:
Image(filename='../../static/images/apps/11A-000_2.png')
[2]:
CSS¶
A custom css file is used to improve the interface of this application. It can be found here.
[3]:
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¶
[4]:
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
import urllib.parse
import webbrowser
import sys
Physical functions¶
This are the functions that have a physical meaning:
get_relative_isothermsget_derivative_y_by_x
[5]:
def get_relative_isotherms(v_range, T_range):
"""This function calculates the theoretical p(v, T) plane
(in reduced coordinates) according to van der Waals
equation of state from a given range of volumes
and tenperatures.
Args:
v_range: An array containing the values of v
(in reduced coordinates)for which the isotherms must be
calculated.\n
T_range: An array containing the values of T
(in reduced coordinates)for which the isotherms must be
calculated.\n
Returns:
isotherms: A list consisted of numpy arrays containing the
pressures of each isotherm.
"""
isotherms = []
for T in T_range:
p_R = []
for v in v_range:
val = (8.0/3.0*T/(v - 1.0/3.0) - 3.0/v**2)
p_R = np.append(p_R, val)
isotherms.append(p_R)
return isotherms
[6]:
def get_derivative_y_by_x(y_values, x_values):
"""This function calculates the derivative an y array
with respect to an x array calculated with the difference quotient.
Args:
y_values: An array containing the values of y.\n
x_values: An array containing the values of x.\n
Returns:
der: An array containing the values of the
derivative of y_values with respect to x_values.
"""
der = []
for i in range(len(x_values)):
x = x_values[i]
y = y_values[i]
d = []
l = np.size(y)
for j in range(1, l):
d.append((y[j] - y[j-1])/(x[j] - x[j-1]))
der.append(d)
return der
Main interface¶
[ ]:
v_values = np.linspace(0.4, 5.0, 500)
T_values = [0.85, 0.9, 0.95, 1.0, 1.05, 1.1, 1.15, 1.2]
p_values = get_relative_isotherms(v_values, T_values)
################################
######TOP BLOCK#################
################################
top_block = widgets.VBox(
[],
layout=widgets.Layout(align_items='center')
)
T_slider = widgets.SelectionSlider(
options= T_values,
value=T_values[0],
description=r'\( T \)',
disabled=False,
continuous_update=True,
orientation='horizontal',
readout=True,
layout = widgets.Layout(
width = '300px',
height = 'auto',
margin='0 0 0 50px'
)
)
T_slider.observe(change_tenperature, 'value')
change_view_button = widgets.ToggleButton(
value=False,
description='Presentation mode (OFF)',
disabled=False,
button_style='',
tooltip='',
icon='desktop',
layout=widgets.Layout(
width='initial',
align_self='center'
)
)
change_view_button.observe(change_view, 'value')
top_block.children = [
change_view_button,
T_slider
]
################################
######BOTTOM BLOCK##############
################################
bottom_block = widgets.HBox(
[],
layout=widgets.Layout(
width='100%',
align_items='center'
)
)
dense_v_values = np.linspace(min(v_values), max(v_values), 10000)
dense_p_values = get_relative_isotherms(dense_v_values, T_values)
dense_v_values_filtered = []
dense_p_values_filtered = []
dense_v_values_inverted = []
dense_p_values_inverted = []
dense_v_values_rounded = []
for i in range(len(T_values)):
i_in_range, = np.where(dense_p_values[i] < 2.0)
dense_v_values_filtered.append(np.take(dense_v_values, i_in_range))
dense_p_values_filtered.append(np.take(dense_p_values[i], i_in_range))
dense_v_values_rounded.append(np.round(dense_v_values_filtered[i], 3))
v_text = widgets.HTML(
value="<p>" + str(dense_v_values_rounded[T_slider.index][i]) + "</p>",
layout = widgets.Layout(
height='auto',
margin='8px 0 0 10px',
width='initial'
)
)
der = get_derivative_y_by_x(dense_p_values_filtered, dense_v_values_filtered)
v_slider = widgets.IntSlider(
min=0,
max=len(der[T_slider.index])-1,
value=0,
description=r'\( v \)',
disabled=False,
continuous_update=True,
orientation='horizontal',
readout=False,
layout = widgets.Layout(width = '75%', height='auto', margin='0 0 0 100px')
)
v_slider.observe(update_tracer, 'value')
v_slider.observe(update_text, 'value')
play = widgets.Play(
interval=1,
value=0,
min=0,
max=v_slider.max,
step=1,
description="Press play",
disabled=False
)
widgets.jslink((play, 'value'), (v_slider, 'value'));
scale_x = bqs.LinearScale(min = min(v_values), max = max(v_values))
scale_y = bqs.LinearScale(min = 0.0, max = 2.0)
color_scale = bqs.ColorScale(
colors = ['#FF0000', '#FFfa00'],
min=min(T_values),
max=max(T_values)
)
axis_x_001 = bqa.Axis(
scale=scale_x,
tick_format='.2f',
tick_style={'font-size': '15px'},
tick_values = [0.5, 2.5, 5.0],
grid_lines = 'none',
grid_color = '#8e8e8e',
label='v',
label_location='middle',
label_style={'stroke': 'black', 'default-size': 35},
label_offset='50px'
)
axis_y_001 = bqa.Axis(
scale=scale_y,
tick_format='.1f',
tick_style={'font-size': '15px'},
tick_values = [0.0, 1, 2],
grid_lines = 'none',
grid_color = '#8e8e8e',
orientation='vertical',
label='p',
label_location='middle',
label_style={'stroke': 'red', 'default_size': 35},
label_offset='50px'
)
axis_color = bqa.ColorAxis(
label = 'T',
scale=color_scale,
tick_format='.2f',
orientation='vertical',
side='right'
)
fig_11A_001 = Figure(
title='p vs v (fixed T, reduced variables)',
marks=[],
axes=[axis_x_001, axis_y_001, axis_color],
animation_duration=0,
legend_location='top-right',
background_style= {'fill': 'white', 'stroke': 'black'},
fig_margin=dict(top=70, bottom=75, left=80, right=100),
toolbar = True,
layout = widgets.Layout(width='85%')
)
lines_11A_001 = bqm.Lines(
x = v_values,
y = p_values,
scales = {'x': scale_x, 'y': scale_y, 'color': color_scale},
opacities = [1.0] + [0.0 for i in range(len(T_values)-1)],
visible = True,
color = T_values,
)
tracer_11A_001 = bqm.Scatter(
name = '',
x = [0.0],
y = [0.0],
scales = {'x': scale_x, 'y': scale_y},
opacity = [1.0],
visible = False,
colors = ['#2807a3'],
)
fig_11A_001.marks = [
lines_11A_001,
tracer_11A_001
]
scale_x_002 = bqs.LinearScale(min = 0.0, max = 2.0)
scale_y_002 = bqs.LinearScale(min = -2.0, max = 2.0)
axis_y_002 = bqa.Axis(
scale=scale_y_002,
tick_format='.2f',#'0.2f',
tick_style={'font-size': '15px'},
tick_values = [-2, -1, 0, 1, 2],
grid_lines = 'none',
grid_color = '#8e8e8e',
orientation='vertical',
label='dp/dv',
label_location='middle',
label_style={'stroke': 'red', 'default_size': 35},
label_offset='50px'
)
fig_11A_002 = Figure(
title='dp/dv vs v (fixed T, reduced variables)',
marks=[],
axes=[axis_x_001, axis_y_002],
animation_duration=0,
legend_location='top-right',
background_style= {'fill': 'white', 'stroke': 'black'},
fig_margin=dict(top=70, bottom=75, left=80, right=100),
toolbar = True,
layout = widgets.Layout(width='85%')
)
lines_11A_002 = bqm.Lines(
x = [0.4],
y = [1.0],
scales = {'x': scale_x, 'y': scale_y_002, 'color':color_scale},
opacities = [1.0],
visible = True,
color = T_values,
)
zero_11A_002 = bqm.Lines(
x = v_values,
y = [0.0 for v in v_values],
scales = {'x': scale_x, 'y': scale_y_002},
opacities = [1.0],
visible = True,
colors = ['#FF0000'],
line_style = 'dotted'
)
fig_11A_002.marks = [
zero_11A_002,
lines_11A_002
]
restart_button = widgets.Button(
description='Clean figure',
disabled=False,
button_style='',
tooltip="",
layout = widgets.Layout(height='auto')
)
restart_button.on_click(restart_derivative)
show_all_button = widgets.Button(
description='Show all derivatives',
disabled=False,
button_style='',
tooltip="",
layout = widgets.Layout(height='auto', width='initial')
)
show_all_button.on_click(show_all_derivatives)
prepare_export_fig_11A_001_button = widgets.Button(
description='Export',
disabled=False,
button_style='',
tooltip='',
)
prepare_export_fig_11A_001_button.on_click(prepare_export)
prepare_export_fig_11A_002_button = widgets.Button(
description='Export',
disabled=False,
button_style='',
tooltip='',
)
prepare_export_fig_11A_002_button.on_click(prepare_export)
slider_box = widgets.HBox([
v_slider, v_text
],
layout=widgets.Layout(
height='auto',
width='100%',
align_items='center'
)
)
bottom_block.children = [
widgets.VBox([
fig_11A_001,
prepare_export_fig_11A_001_button,
slider_box,
play
],
layout=widgets.Layout(
height='auto',
width='50%',
align_items='center'
)
),
widgets.VBox([
fig_11A_002,
prepare_export_fig_11A_002_button,
restart_button,
show_all_button
],
layout=widgets.Layout(
height='auto',
width='50%',
align_items='center'
)
)
]
main_block_11A_000 = widgets.VBox(
[],
layout=widgets.Layout(align_items='center')
)
main_block_11A_000.children = [
top_block,
bottom_block
]
figures = [
fig_11A_001,
fig_11A_002
]
main_block_11A_000