Mathematical analysis of the van der Waals isotherms

Code: #118-000

File: apps/van_der_waals/mathematical_analysis.ipynb

Run it online: Binder

The aim of this notebook is to show the mathematical function of van der Waals isotherms.

Interface

The main interface (main_block_118_000) is divided in two HBox: top_block_118_000 and bottom_block_118_000. top_block_118_000 contains of a bqplot Figure (fig_118_001) and bottom_block_118_000 contains 4 bqplot Figures: fig_118_003, fig_118_004, fig_118_005 and fig_118_006. The slider zoom_slider controls the zoom of fig_118_001.

[1]:

from IPython.display import Image
Image(filename='../../static/images/apps/118-000_1.png')

[1]:
../../_images/apps_van_der_waals_mathematical_analysis_3_0.png 
[2]:

Image(filename='../../static/images/apps/118-000_2.png')

[2]:
../../_images/apps_van_der_waals_mathematical_analysis_4_0.png

CSS

A custom css file is used to improve the interface of this application. It can be found here.

[1]:

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>"))

Packages

[2]:

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:

  • calculate_critic
  • get_absolute_isotherms
  • bar_to_atm
[3]:

def calculate_critic(a, b):

    """
        This function calculates the critic point
        (p_c, v_c, T_c) from given a and b parameters of
        the Van der Waals equation of state for real gases.

        :math:`(P + a \\frac{n^2}{V^2})(V - nb) = nRT`

        :math:`p_c = \\frac{a}{27 b^2}`
        :math:`v_c = 3b`
        :math:`T_c = \\frac{8a}{27 b R}`

   Args:
       a: Term related with the attraction between particles in
       L^2 bar/mol^2.\n
       b: Term related with the volume that is occupied by one
       mole of the molecules in L/mol.\n

   Returns:
       p_c: Critical pressure in bar.\n
       v_c: Critical volume in L/mol.\n
       T_c: Critical tenperature in K.\n

    """

    if b == 0.0:
        return None

    k_B = 1.3806488e-23 #m^2 kg s^-2 K^-1
    N_A = 6.02214129e23
    R = 0.082 * 1.01325 #bar L mol^-1 K^-1

    p_c = a/27.0/(b**2)
    v_c = 3.0*b
    T_c = 8.0*a/27.0/b/R

    return p_c, v_c, T_c

[4]:

def get_absolute_isotherms(a, b, v_values, T_values):
    """This function calculates the theoretical p(v, T) plane
        (in absolute coordinates) according to van der Waals
        equation of state from a given range of volumes
        and tenperatures.

    Args:
        a: Term related with the attraction between particles in
           L^2 bar/mol^2.\n
        b: Term related with the volume that is occupied by one
        mole of the molecules in L/mol.\n
        v_values: An array containing the values of v
        for which the isotherms must be calculated.\n
        T_values: An array containing the values of T for which
        the isotherms must be calculated.\n


    Returns:
        isotherms: A list consisted of numpy arrays containing the
        pressures of each isotherm.
    """
    isotherms = []

    R = 0.082 * 1.01325 #bar L mol^-1 K^-1

    for T in T_values:

        isot = []

        for v in v_values:

            p = R*T/(v - b) - (a/v**2)
            isot = np.append(isot, p)

        isotherms.append(isot)

    return isotherms

[5]:

def bar_to_atm(p_values):
    """This function changes the pressures of an array
    form bars to atm.

    Args:
        p_values: List consisted of pressures in bars.\n

    Returns:
        p_values: List consisted of pressures in atm.\n
    """

    p_values = np.array(p_values) * 0.9869

    return p_values

Main interface

[ ]:

#In this program we are going to use water's parameters
a = 5.536 #L^2 bar mol^-2
b = 0.03049 #L mol^-1

colors = ['#0079c4','#f09205','#21c400', '#850082']

p_c, v_c, T_c = calculate_critic(a, b)

p_c = p_c * 0.9869 #unit change from bar to atm

v_values = np.linspace(-5.0, 5.0, 3000) #L/mol
T_values = [0.9*T_c, 1.0*T_c, 1.1*T_c]

p_values = get_absolute_isotherms(a, b, v_values, T_values)
p_values = bar_to_atm(p_values)

v_values_hd = np.linspace(-20, 20, 10000)
p_values_hd = get_absolute_isotherms(a, b, v_values_hd, T_values)

#####################
######TOP BLOCK######
#####################

top_block_118_000 = widgets.VBox(
    [],
    layout=widgets.Layout(align_items='center')
)

scale_x_118_001 = bqs.LinearScale(min = min(v_values), max = max(v_values))
scale_y_118_001 = bqs.LinearScale(min =-500.0, max = 500.0)

axis_x_118_001 = bqa.Axis(
    scale=scale_x_118_001,
    tick_format='.1f',
    tick_style={'font-size': '15px'},
    num_ticks=5,
    grid_lines = 'none',
    grid_color = '#8e8e8e',
    label='v (L/mol)',
    label_location='middle',
    label_style={'stroke': 'black', 'default-size': 35},
    label_offset='50px'
)

axis_y_118_001 = bqa.Axis(
    scale=scale_y_118_001,
    tick_format='.0f',
    tick_style={'font-size': '15px'},
    num_ticks=5,
    grid_lines = 'none',
    grid_color = '#8e8e8e',
    orientation='vertical',
    label='p (atm)',
    label_location='middle',
    label_style={'stroke': 'red', 'default_size': 35},
    label_offset='50px'
)

fig_118_001 = Figure(
    title='',
    marks=[],
    axes=[axis_x_118_001, axis_y_118_001],
    animation_duration=500,
    legend_location='top-right',
    background_style= {'fill': 'white',  'stroke': 'black'},
    min_aspect_ratio=1.0,
    fig_margin=dict(top=70, bottom=60, left=80, right=30),
    toolbar = True,
    layout = widgets.Layout(),
)

marks = [
    bqm.Lines(
        x = [v_values for elem in p_values],
        y = p_values,
        scales = {'x': scale_x_118_001, 'y': scale_y_118_001},
        opacities = [1.0],
        visible = True,
        colors = colors,
    )
]

fig_118_001.marks = marks

prepare_export_fig_118_001_button = widgets.Button(
    description='Export',
    disabled=False,
    button_style='',
    tooltip='',
    layout=widgets.Layout(
        width='initial',
        align_self='center'
    )
)

prepare_export_fig_118_001_button.on_click(prepare_export)


zoom_slider = widgets.IntSlider(
    value=0,
    min=0,
    max=30,
    step=1,
    description='Zoom:',
    disabled=False,
    continuous_update=True,
    orientation='vertical',
    readout=False,
    readout_format='d',
    layout = widgets.Layout(margin='35px 0 0 0', height='80%')
)

zoom_slider.observe(update_scales, 'value')


# Calculate the values of the scales
x_min = np.linspace(scale_x_118_001.min, 0.0, zoom_slider.max+1)
x_max = np.linspace(scale_x_118_001.max, 5.0*v_c, zoom_slider.max+1)

y_min = np.linspace(scale_y_118_001.min, 0.0, zoom_slider.max+1)
y_max = np.linspace(scale_y_118_001.max, 456.0, zoom_slider.max+1)


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_118_000.children = [
    change_view_button,
    widgets.HBox([
        widgets.VBox([
            fig_118_001,
            prepare_export_fig_118_001_button,
        ]),
        zoom_slider
    ])
]


#####################
####BOTTOM BLOCK#####
#####################

bottom_block_118_000 = widgets.HBox(
    [],
    layout=widgets.Layout(
        height='300px',
        width='100%'
    )
)

scale_x_118_003 = bqs.LinearScale(
    min = scale_x_118_001.min,
    max = scale_x_118_001.max
)

scale_y_118_003 = bqs.LinearScale(
    min = scale_y_118_001.min,
    max = scale_y_118_001.max
)

axis_x_118_003 = bqa.Axis(
    scale=scale_x_118_003,
    tick_format='.0f',
    tick_style={'font-size': '15px'},
    num_ticks=5,
    grid_lines = 'none',
    grid_color = '#8e8e8e',
    label='v (L/mol)',
    label_location='middle',
    label_style={'stroke': 'black', 'default-size': 35},
    label_offset='50px'
)

axis_y_118_003 = bqa.Axis(
    scale=scale_y_118_003,
    tick_format='.0f',
    tick_style={'font-size': '15px'},
    num_ticks=5,
    grid_lines = 'none',
    grid_color = '#8e8e8e',
    orientation='vertical',
    label='p (atm)',
    label_location='middle',
    label_style={'stroke': 'red', 'default_size': 35},
    label_offset='50px'
)

marks = [
    bqm.Lines(
        x = [v_values for elem in p_values],
        y = p_values,
        scales = {'x': scale_x_118_003, 'y': scale_y_118_003},
        opacities = [1.0],
        visible = True,
        colors = colors
    )
]

fig_118_003 = Figure(
    title='',
    marks=marks,
    axes=[axis_x_118_003, axis_y_118_003],
    animation_duration=0,
    legend_location='top-right',
    background_style= {'fill': 'white',  'stroke': 'black'},
    min_aspect_ratio=1.0,
    fig_margin=dict(top=50, bottom=60, left=80, right=30),
    toolbar = False,
    layout = widgets.Layout(height='90%', width='95%')
)

scale_x_118_004 = bqs.LinearScale(min = -2.0, max = 2.0)
scale_y_118_004 = bqs.LinearScale(min = -2.2*p_c, max = 2.2*p_c)

axis_x_118_004 = bqa.Axis(
    scale=scale_x_118_004,
    tick_format='.0f',
    tick_style={'font-size': '15px'},
    num_ticks=5,
    grid_lines = 'none',
    grid_color = '#8e8e8e',
    label='v (L/mol)',
    label_location='middle',
    label_style={'stroke': 'black', 'default-size': 35},
    label_offset='50px'
)

axis_y_118_004 = bqa.Axis(
    scale=scale_y_118_004,
    tick_format='.0f',
    tick_style={'font-size': '15px'},
    tick_values = [-500, -250, 0, 250, 500],
    grid_lines = 'none',
    grid_color = '#8e8e8e',
    orientation='vertical',
    label='p (atm)',
    label_location='middle',
    label_style={'stroke': 'red', 'default_size': 35},
    label_offset='50px'
)

marks = [
    bqm.Lines(
        x = [v_values_hd for elem in p_values_hd],
        y = p_values_hd,
        scales = {'x': scale_x_118_004, 'y': scale_y_118_004},
        opacities = [1.0],
        visible = True,
        colors = colors,
    )
]

fig_118_004 = Figure(
    title='',
    marks=marks,
    axes=[axis_x_118_004, axis_y_118_004],
    animation_duration=0,
    legend_location='top-right',
    background_style= {'fill': 'white',  'stroke': 'black'},
    min_aspect_ratio=1.0,
    fig_margin=dict(top=50, bottom=60, left=80, right=30),
    toolbar = True,
    layout = widgets.Layout(height='90%', width='95%')
)



scale_x_118_005 = bqs.LinearScale(min = 0.0, max = 2.0)
scale_y_118_005 = bqs.LinearScale(min = 0.0, max = 2.2*p_c)

axis_x_118_005 = bqa.Axis(
    scale=scale_x_118_005,
    tick_format='.1f',
    tick_style={'font-size': '15px'},
    num_ticks=5,
    grid_lines = 'none',
    grid_color = '#8e8e8e',
    label='v (L/mol)',
    label_location='middle',
    label_style={'stroke': 'black', 'default-size': 35},
    label_offset='50px'
)

axis_y_118_005 = bqa.Axis(
    scale=scale_y_118_005,
    tick_format='.0f',
    tick_style={'font-size': '15px'},
    tick_values = [0, 150, 300, 450],
    grid_lines = 'none',
    grid_color = '#8e8e8e',
    orientation='vertical',
    label='p (atm)',
    label_location='middle',
    label_style={'stroke': 'red', 'default_size': 35},
    label_offset='50px'
)

marks = [
    bqm.Lines(
        x = [v_values_hd for elem in p_values_hd],
        y = p_values_hd,
        scales = {'x': scale_x_118_005, 'y': scale_y_118_005},
        opacities = [1.0],
        visible = True,
        colors = colors,
    )
]

fig_118_005 = Figure(
    title='',
    marks=marks,
    axes=[axis_x_118_005, axis_y_118_005],
    animation_duration=0,
    legend_location='top-right',
    background_style= {'fill': 'white',  'stroke': 'black'},
    min_aspect_ratio=1.0,
    fig_margin=dict(top=50, bottom=60, left=80, right=30),
    toolbar = True,
    layout = widgets.Layout(height='90%', width='95%')
)

scale_x_118_006 = bqs.LinearScale(min = 0.5*v_c, max = 5.0*v_c)
scale_y_118_006 = bqs.LinearScale(min = 0.0, max = 2.0*p_c)

axis_x_118_006 = bqa.Axis(
    scale=scale_x_118_006,
    tick_format='.2f',
    tick_style={'font-size': '15px'},
    num_ticks=5,
    grid_lines = 'none',
    grid_color = '#8e8e8e',
    label='v (L/mol)',
    label_location='middle',
    label_style={'stroke': 'black', 'default-size': 35},
    label_offset='50px'
)

axis_y_118_006 = bqa.Axis(
    scale=scale_y_118_006,
    tick_format='.0f',
    tick_style={'font-size': '15px'},
    tick_values = [0, 100, 200, 300, 400],
    grid_lines = 'none',
    grid_color = '#8e8e8e',
    orientation='vertical',
    label='p (atm)',
    label_location='middle',
    label_style={'stroke': 'red', 'default_size': 35},
    label_offset='50px'
)

marks = [
    bqm.Lines(
        x = [v_values_hd for elem in p_values_hd],
        y = p_values_hd,
        scales = {'x': scale_x_118_006, 'y': scale_y_118_006},
        opacities = [1.0],
        visible = True,
        colors = colors,
    )
]

fig_118_006 = Figure(
    title='',
    marks=marks,
    axes=[axis_x_118_006, axis_y_118_006],
    animation_duration=0,
    legend_location='top-right',
    background_style= {'fill': 'white',  'stroke': 'black'},
    min_aspect_ratio=1.0,
    fig_margin=dict(top=50, bottom=60, left=80, right=30),
    toolbar = True,
    layout = widgets.Layout(height='90%', width='95%')
)

prepare_export_fig_118_003_button = widgets.Button(
    description='Export',
    disabled=False,
    button_style='',
    tooltip='',
    layout=widgets.Layout(
        width='initial',
        align_self='center'
    )
)

prepare_export_fig_118_003_button.on_click(prepare_export)

prepare_export_fig_118_004_button = widgets.Button(
    description='Export',
    disabled=False,
    button_style='',
    tooltip='',
    layout=widgets.Layout(
        width='initial',
        align_self='center'
    )
)

prepare_export_fig_118_004_button.on_click(prepare_export)

prepare_export_fig_118_005_button = widgets.Button(
    description='Export',
    disabled=False,
    button_style='',
    tooltip='',
    layout=widgets.Layout(
        width='initial',
        align_self='center'
    )
)

prepare_export_fig_118_005_button.on_click(prepare_export)

prepare_export_fig_118_006_button = widgets.Button(
    description='Export',
    disabled=False,
    button_style='',
    tooltip='',
    layout=widgets.Layout(
        width='initial',
        align_self='center'
    )
)

prepare_export_fig_118_006_button.on_click(prepare_export)

bottom_block_118_000.children = [
    widgets.VBox([
        fig_118_003,
        prepare_export_fig_118_003_button,
        ],
        layout=widgets.Layout(
            width='25%',
        )
    ),
    widgets.VBox([
        fig_118_004,
        prepare_export_fig_118_004_button,
    ],
        layout=widgets.Layout(
            width='25%',
        )
    ),
    widgets.VBox([
        fig_118_005,
        prepare_export_fig_118_005_button,
    ],
        layout=widgets.Layout(
            width='25%',
        )
    ),
    widgets.VBox([
        fig_118_006,
        prepare_export_fig_118_006_button,
    ],
        layout=widgets.Layout(
            width='25%',
        )
    ),
]

#####################
####MAIN BLOCK#####
#####################

main_block_118_000 = widgets.VBox(
    [],
    layout=widgets.Layout(align_items='center')
)

main_block_118_000.children = [
    top_block_118_000,
    bottom_block_118_000
]

figures = [
    fig_118_001,
    fig_118_003,
    fig_118_004,
    fig_118_005,
    fig_118_006,
]

main_block_118_000