Demonstration of the Invalidity of the Mpemba Effect


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<front>
<journal-meta>
<journal-id journal-id-type="publisher-id">IJAERD</journal-id>
<journal-title-group>
<journal-title>International Journal of Applied Engineering Research and Development</journal-title>
</journal-title-group>
<issn pub-type="ppub">2250-1584</issn>
<issn pub-type="epub">2278-9383</issn>
<publisher>
<publisher-name>Trans Stellar Journal Publications Research Consultancy</publisher-name>
<publisher-loc>India</publisher-loc>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="publisher-id">IJAERD-8-13</article-id>
<article-id pub-id-type="paper-id">IJAERDDEC20182</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Original Article</subject>
</subj-group>
</article-categories>
<title-group>
<article-title>DEMONSTRATION OF THE INVALIDITY OF THE MPEMBA EFFECT</article-title></title-group>
<contrib-group content-type="article">
<contrib contrib-type="author">
<name>
<surname>VERDE</surname>
<given-names>GIUSEPPE</given-names>
</name>
<role>Student</role>
</contrib>
<aff id="aff">Parthenope University of Naples, Italy</aff>
</contrib-group>
<pub-date pub-type="ppub">
<season>Oct</season>
<year>2018</year>
</pub-date>
<volume>8</volume>
<issue>2</issue>
<fpage>13</fpage>
<lpage>22</lpage>
<history>
<date date-type="received">
<day>07</day>
<month>Aug</month>
<year>2018</year>
</date>
<date date-type="accepted">
<day>27</day>
<month>Aug</month>
<year>2018</year>
</date>
<date date-type="published">
<day>21</day>
<month>Sep</month>
<year>2018</year>
</date>
</history>
<permissions>
<copyright-statement>&#x00A9; TJPRC Pvt. Ltd.</copyright-statement>
<copyright-year>2018</copyright-year>
<copyright-holder>www.tjprc.org</copyright-holder>
</permissions>
<abstract>
<p>Demonstration of the invalidity of mpemba effect with thermodynamic processes and its operating method are described. The mpemba effect is a process, in which hot water can freeze faster than cold water. This assertion is counter-intuitive. Indeed, the mpemba effect breaches fundamental thermodynamic laws. In fact, that there is no evidence meaningful of the validity of the mpemba effect &#x005B;<xref ref-type="bibr" rid="ref1">1</xref>&#x005D;:the phenomenon, when taken to mean &#x201C;hot water freezes faster than cold&#x201D;, is difficult to reproduce or confirm because this statement is ill-defined; although the mpemba effect is real, it is unclear whether the explanation is trivial or enlightening. However, with these definitions, this effect has never been clear. On the other hand, there are many the circumstances in which the effect is not observed. The main references to the effect go back at least to the writings of aristotle &#x005B;<xref ref-type="bibr" rid="ref2">2</xref>&#x005D;, but the author shows that this effect is mistaken and shows the available evidence for negation of the mpemba effect and carry out experiments by cooling water in carefully controlled conditions. A mathematical demonstration has been presented, that violates the physical principle, coming to a contradiction of thermodynamic laws. The demonstration would show therefore, the mpemba effect cannot occur, and it does not exist if the initial parameters are correctly equivalents. The required vast multidimensional array of experiments explains why the effect is not yet understood and demonstrated.</p>
<p>Therefore, it is necessary to verify all the previous/historical relationships are the result of a bad experimental technique, since evidently it started from incorrect assumptions. It is important and necessary to stress that, in order to validate the thesis: all conditions must be the same at any time.</p>
</abstract>
<kwd-group>
<kwd>Thermodynamics</kwd>
<kwd>Mpemba</kwd>
<kwd>Coldest Water</kwd>
<kwd>Hottest Water</kwd>
</kwd-group>
<counts>
<fig-count count="4"/>
<table-count count="1"/>
<ref-count count="40"/>
<page-count count="10"/>
</counts>
</article-meta>
</front>
<body>
<sec id="sec1-1" sec-type="intro">
<title>INTRODUCTION</title>
<p>To describe this phenomenon, in the past, some explanations had been proposed. However, the following observations studied are not sufficient to resolve the effect, including:</p>
<list list-type="bullet">
<list-item><p><bold>Evaporation</bold>: the evaporation of the hottest water reduces the mass of the water to be frozen &#x005B;<xref ref-type="bibr" rid="ref3">3</xref>&#x005D;.</p></list-item>
</list>
<p>But, I will show that this thesis is not valid for the experiment observed. The reason is very simple: the evaporation of water would cause a decrease in mass. Therefore, the experiment cannot be considered valid, since the initial characteristics have changed. System's mass cannot change, so quantity cannot be added nor removed.</p>
<p>Anyway, if the mass differences are present and considered valid for the experiment observed, they are negligible, because don&#x2019;t significantly change the time needed for water to freeze.</p>
<list list-type="bullet">
<list-item><p><bold>Convection</bold>: Accelerating heat transfers. In the warmer water, the convention motion improves the heat exchange with the external environment &#x005B;<xref ref-type="bibr" rid="ref4">4</xref>&#x005D;. Higher convection in the warmer water may spread ice crystals around faster. This thesis, if supposed valid, has a limit.</p></list-item>
</list>
<p>It is considered that:</p>
<p><italic>T<sub>h</sub></italic> = hottest water temperature.</p>
<p><italic>T<sub>c</sub></italic> = coldest water temperature.</p>
<p><italic>t<sub>h</sub></italic> = freezing time for the hottest water.</p>
<p><italic>t<sub>c</sub></italic> = freezing time for the coldest water.</p>
<p>In this experiment, water is cooled to 0&#x00B0;C for two water samples. It is supposed that hottest water takes a time to switch from temperature <italic>T<sub>h</sub></italic> to the <italic>T<sub>c</sub></italic> temperature. Meantime, the coldest water assumes a temperature <italic>T</italic> &#x003C; <italic>T<sub>c</sub></italic>. From the instant of time <italic>t</italic><sub>1</sub>, the hottest water will proceed at the same speed as the coldest water, but at a higher temperature. Therefore, the hottest water will reach the temperature of 0&#x00B0;C in a time <italic>t<sub>h</sub></italic> &#x003E; <italic>t<sub>c</sub></italic>. <xref ref-type="fig" rid="fig1">Figure 1</xref> shows the thermal effect explained through an asymptotic representation.</p>
<fig id="fig1">
<label>Figure 1</label>
<caption>
<p><bold>A Graph of Freezing Rates for the Two Water Samples</bold></p>
</caption>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/fig_1.jpg"/>
</fig>
<list list-type="bullet">
<list-item><p><bold>Frost</bold>: it would have insulating effects. The coldest water would tend to freeze from the top, reducing further heat loss, while the hottest water would tend to freeze from the bottom.</p>
<p>Similarly, the previous demonstration for the convection process can also be used in this case.</p>
</list-item>
<list-item><p><bold>Thermal Conductivity</bold>: the hottest water can melt through a layer of frost that acts as an insulator, allowing the water to come into direct contact with a lower layer much colder. Therefore, the hottest water then cools much more quickly from this point on.</p></list-item>
</list>
<p>But also in this case, it is shown that this thesis is not valid for the experiment: the hottest water absorbs more thermal energy. Therefore, the experiment cannot be considered valid, since the initial conditions of the observations are different.</p>
<p>Finally, all the other cases are attributable to the previous demonstration for the convection process, among which:</p>
<list list-type="bullet">
<list-item><p><bold>Dissolved Gases</bold>: coldest water would contain more dissolved gases than the hottest water. This phenomenon would change the properties of the water with respect to convection currents.</p></list-item>
<list-item><p><bold>Hydrogen Bonding</bold>: in warm water, hydrogen bonding is weaker.</p></list-item>
<list-item><p><bold>Crystallization</bold>: The hottest water might be responsible for the faster crystallization due to relatively higher population of hexamer states &#x005B;<xref ref-type="bibr" rid="ref5">5</xref>&#x005D;.</p></list-item>
<list-item><p><bold>Distribution Functions</bold>: from the Maxwell-Boltzmann distribution showing up in gases &#x005B;<xref ref-type="bibr" rid="ref6">6</xref>&#x005D;.</p></list-item>
</list>
<p>If these phenomena are present, they are considered valid in a limited time for the experiment observed.</p>
<p>Therefore, it is clear that the explanation is trivial. Although the Mpemba effect could be real, it is only the consequence of obvious characteristics.</p>
</sec>
<sec id="sec1-2">
<title>THEORETICAL REFERENCES</title>
<p>Sensible heat is defined as heat exchanged by a body or thermodynamic system, which produce a decrease in the temperature difference, but leaving unchanged the other thermodynamic parameters, such as volume and pressure&#x005B;<xref ref-type="bibr" rid="ref7">7</xref>&#x005D; &#x005B;<xref ref-type="bibr" rid="ref8">8</xref>&#x005D; &#x005B;<xref ref-type="bibr" rid="ref9">9</xref>&#x005D; &#x005B;<xref ref-type="bibr" rid="ref10">10</xref>&#x005D;.</p>
<p>The sensible heat (<italic>Q<sub>s</sub></italic>) of a thermal process may be calculated as the product of the thermodynamic system mass (m) with its specific heat capacity (c) and the change in temperature (&#x2206;T):</p>
<p>Q<sub>s</sub> = <italic>mc</italic>&#x2206;T</p>
<p>As a form of energy, heat (Q) has the unit joule (J) in the International System of Units (SI), whereas the standard International System of Units, (SI) unit of mass is the kilogram (kg). Heat capacity or thermal capacity is defined as the ratio of the heat added to or removed from a thermodynamic system to the resulting temperature change. The unit of heat capacity is joule per kelvin in the International System of Units (SI):</p>
<p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn1.png"/></p>
<p>The specific heat capacity or specific heat is the heat capacity per unit mass of a thermodynamic system:</p>
<p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn2.png"/></p>
<p>In fact, in science and engineering, thermodynamic properties are often defined by the specific term &#x005B;<xref ref-type="bibr" rid="ref11">11</xref>&#x005D; &#x005B;<xref ref-type="bibr" rid="ref12">12</xref>&#x005D;.</p>
<p>The term <italic>Q<sub>s</sub></italic> is used differently with latent heat (<italic>Q<sub>L</sub></italic>). Latent heat is the thermal energy released or absorbed by a thermodynamic system during an isothermal and isobaric process, involved respectively in phase changes at a specified temperature and pressure. It is defined as the latent heat of fusion for a substance condensing and latent heat of vaporization for a substance vaporizing &#x005B;<xref ref-type="bibr" rid="ref13">13</xref>&#x005D; &#x005B;<xref ref-type="bibr" rid="ref14">14</xref>&#x005D;. From this definition, the latent heat (<italic>Q<sub>L</sub></italic>) for a given mass (m) of a considered thermodynamic system substance, necessary for the phase transition is:</p>
<p><italic>Q<sub>L</sub></italic> = <italic>m&#x03BB;</italic></p>
<p>Where, &#x03BB; is the specific latent heat for a particular substance:</p>
<p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn3.png"/></p>
<p>In the scientific literature, there are commonly quoted/tabulated and defined specific terms, latent heat of fusion (<italic>&#x03BB;<sub>f</sub></italic>) and the specific latent heat of vaporization (<italic>&#x03BB;<sub>v</sub></italic>) to describe a substance of a thermodynamic system.</p>
<p>Sensible heat and latent heat describe exchanges of heat under conditions prefixed/specified. Therefore, they aren&#x2019;t special forms of energy and aren&#x2019;t described a property of one system, or contained within it, because the heat is defined as the energy transferred spontaneously from a hottest to a coldest thermodinamyc system &#x005B;<xref ref-type="bibr" rid="ref15">15</xref>&#x005D; &#x005B;<xref ref-type="bibr" rid="ref16">16</xref>&#x005D;.</p>
<p>Instead, thermal power (Q&#x0307;) is the amount of heat/thermal energy (Q) transferred per unit time (t). Thermal power (equivalently <italic>P<sub>t</sub></italic>), as a function of time, is the rate at which work is done, so can be expressed by this equation:</p>
<p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn4.png"/></p>
<p>The standard unit for the rate of heat transferred is the watt (W), defined as one joule per second:</p>
<p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn5.png"/></p>
<p>Mpemba effect</p>
<p>The effect is named after Erasto Mpemba, described it in 1963 &#x005B;<xref ref-type="bibr" rid="ref17">17</xref>&#x005D; &#x005B;<xref ref-type="bibr" rid="ref18">18</xref>&#x005D;. The phenomenon is taken to mean &#x201C;hot water freezes faster than cold&#x201D;, is impossible, if the conditions of the observations are identical, as it would violate the first or second law of thermodynamics.</p>
</sec>
<sec id="sec1-3">
<title>THOUGHT EXPERIMENT</title>
<p>It is worth mentioning some hypothesis/principles, for the purpose of thinking through its consequences. The common goal of this thought experiment is to explore the potential consequences of the effect in question, calculated theoretically applying the laws of physics, and carried out to support, refute and validate observations of this research.</p>
<p>Fundamental Proposition: If you take two similar containers with equal volumes of water, one hottest and the other coldest, and put them into a freezer, the coldest water freezes first (<italic>T<sub>f</sub></italic> = 273, 15 K).</p>
</sec>
<sec id="sec1-4">
<title>DEMONSTRATION</title>
<p>&#x005B;<xref ref-type="bibr" rid="ref19">19</xref>&#x005D; New Scientist recommends starting the experiment with containers/glass at: 35&#x00B0;C for hottest water temperature (<italic>T<sub>h</sub></italic>) and 5 &#x00B0;C for coldest water temperature (<italic>T<sub>c</sub></italic>), to maximize the Mpemba effect.</p>
<p>It is highlighted that the hottest water has the following properties:</p>
<p>Temperature (T) = 308,15 <italic>K</italic>; mass (m) = 137,50 <italic>g</italic> = 0,1375 <italic>kg</italic>; specific heat capacity <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn6.png"/>; density <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn7.png"/>; specific volume <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn8.png"/>; volume (V) = 138,30 <italic>ml</italic>.</p>
<p>From these properties, it is possible to calculate the specific heat:</p>
<p><italic>Q<sub>s<sub>h</sub></sub> = mc&#x2206;T<sub>s<sub>h</sub></sub></italic> = 20,13 <italic>kJ</italic></p>
<p>The heat capacity is often affected by many of the state variables that describe the thermodynamic system, among which the temperature, pressure and volume. So, there are actually different measurements of heat capacity. The most commonly used methods for measurement are to hold the thermodynamic system either at constant pressure or at constant volume. In this specific case, the used method for measurement is at constant pressure. Therefore, at constant pressure:</p>
<p><italic>Q<sub>s<sub>h</sub></sub> = m&#x2206;h<sub>s<sub>h</sub></sub></italic></p>
<p>Enthalpy of a thermodynamic system substance describe the transfer of energy in many forms i. e. the heat &#x005B;<xref ref-type="bibr" rid="ref20">20</xref>&#x005D;.</p>
<p>The enthalpy (H) of a thermodynamic system is equal to the sum of system's internal energy with the product of its volume and pressure &#x005B;<xref ref-type="bibr" rid="ref21">21</xref>&#x005D; &#x005B;<xref ref-type="bibr" rid="ref22">22</xref>&#x005D;.</p>
<p>For isobaric processes, the heat absorbed or released equals the change of enthalpy.</p>
<p>The unit of measurement for enthalpy in the International System of Units (SI) is the joule (J). Whereas, The SI unit for specific enthalpy (h) is joule per kilogram:</p>
<p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn9.png"/></p>
<p>It is worth mentioning/looking at the thermodynamic process using the p-h diagram in <xref ref-type="fig" rid="fig2">Figure 2</xref>:</p>
<fig id="fig2">
<label>Figure 2</label>
<caption>
<p><bold>p-h Diagram of Freezing for the Hottest Water</bold></p>
</caption>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/fig_2.jpg"/>
</fig>
<p>With: <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn10.png"/></p>
<p>It is highlighted that the coldest water has the following properties:</p>
<p>Temperature (T) = 278, 15 K; mass (m) = 137,50 <italic>g</italic> = 0,1375 <italic>kg</italic>; specific heat capacity <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn11.png"/>; density <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn12.png"/>; specific volume <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn13.png"/>; volume (V) = 137,50 <italic>ml</italic>.</p>
<p>From these properties, it is possible to calculate the specific heat:</p>
<p>Q<sub>s<sub>c</sub></sub> = <italic>mc&#x2206;T<sub>s<sub>c</sub></sub></italic> = 2,89 <italic>kJ</italic></p>
<p>Also in this case, the used method for measurement is at constant pressure:</p>
<p>Q<sub>s<sub>c</sub></sub> = <italic>mc&#x2206;h<sub>s<sub>c</sub></sub></italic></p>
<p>It is worth mentioning/looking at the thermodynamic process using the p-h diagram in <xref ref-type="fig" rid="fig3">Figure 3</xref>:</p>
<fig id="fig3">
<label>Figure 3</label>
<caption>
<p><bold>p-h Diagram of Freezing for the Coldest Water</bold></p>
</caption>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/fig_3.jpg"/>
</fig>
<p>With: <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn14.png"/></p>
<p>(Q&#x0307;) This means that the change in enthalpy under such conditions is lower. Enthalpies for chemical substances at constant pressure usually refer to standard state: most commonly 1 bar pressure.</p>
<p>Cooling capacity is the measure of a cooling system's ability to remove heat &#x005B;<xref ref-type="bibr" rid="ref23">23</xref>&#x005D;. The SI units are watts (W). The basic SI unit&#x2019;s equation for deriving cooling capacity is of the form:</p>
<p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn19.png"/></p>
<p>(m&#x0307;) Mass flow rate is the mass of a substance, which passes per unit of time. Its unit is kilogram per second in SI units. Sometimes, mass flow rate is termed mass flux or mass current &#x005B;<xref ref-type="bibr" rid="ref24">24</xref>&#x005D;.</p>
<p>It is supposed that the two containers with equal volumes of water, one hottest and the other coldest, are placed in a freezer with a predetermined cooling capacity. The cooling time (t) is measured in seconds, and is determined by the following equation:</p>
<p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn15.png"/></p>
<p>It is highlighted that:</p>
<p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn20.png"/></p>
<p>With:</p>
<p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn16.png"/></p>
<p>Therefore, as it turns out, the coldest water freezes first, this is what the author was trying to demonstrate.</p>
<p>Indeed, when <italic>T<sub>h</sub></italic> reach <italic>T<sub>c</sub></italic>:</p>
<p><italic>T<sub>c</sub></italic> = <italic>T<sub>h</sub> &#x2212; &#x2206;T</italic> If instead absurdly, let us assume that the Mpemba effect is valid, then:</p>
<p><italic>t<sub>s<sub>h</sub></sub></italic> &#x003C; <italic>t<sub>s<sub>c</sub></sub></italic></p>
<p>Therefore, it is worth to start from the first law of thermodynamics for closed systems, for an infinitesimal process:</p>
<p><italic>dU = &#x03B4;Q &#x2212; &#x03B4;L</italic></p>
<p>Where, U is internal energy, L is work and Q is heat &#x005B;<xref ref-type="bibr" rid="ref25">25</xref>&#x005D;. Pressure (p)-volume (V) work by the closed system is defined as:</p>
<p><italic>dU = &#x03B4;Q &#x2212; pdv</italic></p>
<p>The first law of thermodynamics is often is expressed in terms of enthalpy:</p>
<p><italic>dH = &#x03B4;Q + Vd<sub>p</sub></italic></p>
<p>For an isobaric transformation (constant pressure), the first law of thermodynamics is reduced to:</p>
<p><italic>&#x03B4;Q = dH</italic></p>
<p>&#x005B;<xref ref-type="bibr" rid="ref26">26</xref>&#x005D; &#x005B;<xref ref-type="bibr" rid="ref27">27</xref>&#x005D; it is remembered that the enthalpy of a thermodynamic system is defined as:</p>
<p><italic>H = U + pV</italic></p>
<p>About this, assuming that <italic>t<sub>s<sub>h</sub></sub></italic> &#x003C; <italic>t<sub>s<sub>c</sub></sub></italic> then:</p>
<p><italic>Q<sub>s<sub>h</sub></sub></italic> &#x003C; <italic>Q<sub>s<sub>C</sub></sub></italic> &#x21D2; <italic>&#x2206;h<sub>s<sub>h</sub></sub></italic> &#x003C; <italic>&#x2206;h<sub>s<sub>c</sub></sub></italic> &#x21D2; <italic>&#x2206;T<sub>s<sub>h</sub></sub></italic> &#x003C; <italic>&#x2206;T<sub>s<sub>c</sub></sub></italic></p>
<p>The final result is an absurd, since the initial hypothesis was:</p>
<p><italic>&#x2206;T<sub>s<sub>h</sub></sub></italic> &#x003E; <italic>&#x2206;T<sub>s<sub>c</sub></sub></italic></p>
<p>As a matter of fact, the Mpemba effect violates the fundamental thermodynamic laws &#x005B;<xref ref-type="bibr" rid="ref29">29</xref>&#x005D; &#x005B;<xref ref-type="bibr" rid="ref30">30</xref>&#x005D; &#x005B;<xref ref-type="bibr" rid="ref31">31</xref>&#x005D;, because:</p>
<p>(<italic>h</italic><sub>1</sub> &#x2212; <italic>h</italic><sub>2</sub>)<sub>sh</sub> &#x003C; <italic>c&#x2206;T<sub>s<sub>h</sub></sub></italic> &#x21D2; <italic>&#x2206;S<sub>sh</sub></italic> &#x003C; 0</p>
<p><italic>&#x2206;S<sub>sh</sub></italic> introduces the measurement of entropy change &#x005B;<xref ref-type="bibr" rid="ref32">32</xref>&#x005D;. Thermodynamic Entropy change quantifies the entity of some thermal changes, such as heat transfer between systems.</p>
<p>In the International System Unit, the thermodynamic entropy (S) have dimension of energy divided by temperature, and has a unit of joules per kelvin:</p>
<p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn1.png"/></p>
<p>The closed systems tend to increasing entropy, that correspond to irreversible change of energy expended as waste heat, limiting the amount of work a system can do &#x005B;<xref ref-type="bibr" rid="ref33">33</xref>&#x005D;&#x005B;<xref ref-type="bibr" rid="ref34">34</xref>&#x005D;&#x005B;<xref ref-type="bibr" rid="ref35">35</xref>&#x005D;&#x005B;<xref ref-type="bibr" rid="ref36">36</xref>&#x005D;. Similarly, for isolated systems, entropy never decreases. This fact has several important consequences in physic science: first, it prohibits free energy machines, defined as &#x201C;perpetual motion"; and second, it implies the arrow of entropy is directly proportional to the arrow of time. This fact leads to an equation/relation is known as the fundamental thermodynamic relation &#x005B;<xref ref-type="bibr" rid="ref37">37</xref>&#x005D;.</p>
<p>This relation is: <italic>dU = T dS &#x2212; pdV</italic></p>
</sec>
<sec id="sec1-5" sec-type="results">
<title>EXPERIMENTAL RESULTS</title>
<p>To verify the experiment, carried out a procedure to support, refute and validate the hypothesis/thesis, previously demonstrated. Experiments vary greatly in goal and scale, but always rely on repeatable procedure and logical analysis of the results. To prove experimentally the invalidity of the Mpemba effect, the experiment begins with containers/glass at: 35&#x00B0;Cfor hottest water temperature (<italic>T<sub>h</sub></italic>) and 5 &#x00B0;C for coldest water temperature (<italic>T<sub>c</sub></italic>). The experiment should maximize the Mpemba effect &#x005B;<xref ref-type="bibr" rid="ref19">19</xref>&#x005D;. On the contrary, the experimental results will show that the Mpemba effect doesn&#x2019;t occur. The following data were obtained in <xref ref-type="fig" rid="fig4">Figure 4</xref>:</p>
<fig id="fig4">
<label>Figure 4</label>
<caption>
<p><bold>Cooling of Water at 5&#x00B0;C and 35 &#x00B0;C in a Glass</bold></p>
</caption>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/fig_4.jpg"/>
</fig>
<p>Moreover, the experiment was reproduced in n=10 different freezers, with few variations. The materials and methods used are already known &#x005B;<xref ref-type="bibr" rid="ref38">38</xref>&#x005D;. But the result obtained is different, in accordance with the laws of thermodynamics. Some results were obtained in <xref ref-type="table" rid="tab1">table 1</xref>, observing 3 intervals of 10 minutes:</p>
<table-wrap id="tab1">
<label>Table 1</label>
<caption>
<p><bold>Cooling Time of Water at 5&#x00B0;C and 35 &#x00B0;C in a Glass</bold></p>
</caption>
<table frame="box" rules="all">
<thead>
<tr>
<th align="center" valign="middle">&#x00A0;</th>
<th align="center" valign="middle"><italic>t</italic><sub>1</sub> = 600 <italic>s</italic></th>
<th align="center" valign="middle"><italic>t</italic><sub>2</sub> = 1200 <italic>s</italic></th>
<th align="center" valign="middle"><italic>t</italic><sub>3</sub> = 1800 s</th>
</tr>
</thead>
<tbody>
<tr>
<td align="center" valign="middle"><italic>T<sub>h</sub></italic></td>
<td align="center" valign="middle">20,96 &#x00B0;C</td>
<td align="center" valign="middle">12,55 &#x00B0;C</td>
<td align="center" valign="middle">7,51 &#x00B0;C</td>
</tr>
<tr>
<td align="center" valign="middle"><italic>T<sub>c</sub></italic></td>
<td align="center" valign="middle">1,95 &#x00B0;C</td>
<td align="center" valign="middle">1,1 &#x00B0;C</td>
<td align="center" valign="middle">0 &#x00B0;C</td>
</tr>
</tbody>
</table>
</table-wrap>
<p>As to prove that the coldest water freezes first:</p>
<p><italic>t<sub>s<sub>h</sub></sub></italic> &#x003E; <italic>t<sub>s<sub>c</sub></sub></italic></p>
<p>Similarly, for an infinitesimal process, if:</p>
<p><italic>T<sub>h</sub></italic> &#x2212; <italic>T<sub>c</sub></italic> = <italic>dT</italic></p>
<p>Then:</p>
<p><italic>t<sub>s<sub>h</sub></sub></italic> &#x2212; <italic>t<sub>s<sub>c</sub></sub></italic> = <italic>dt</italic></p>
<p>The previous solutions/results have a graphical trend of the exponential function (positive semi-axis) when the base b is between 0 - 1:</p>
<p><italic>y = c b<sup>x</sup></italic></p>
<p>In which, the argument x occurs as an exponent and c occurs as a constant, which depends on the environmental conditions, initial observations, properties and many other factors.</p>
</sec>
<sec id="sec1-6" sec-type="conclusions">
<title>CONCLUSIONS</title>
<p>It has been shown that the effect: &#x201C;hot water can freeze faster than cold water&#x201D; is not a real/correct effect &#x005B;<xref ref-type="bibr" rid="ref39">39</xref>&#x005D;, but it is only the result of trivial operations. The laws of physics are universal and related only to the considered reference system. Moreover, &#x005B;<xref ref-type="bibr" rid="ref40">40</xref>&#x005D; the definition/formulation: &#x201C;there exists a set of initial parameters, and a pair of temperatures, such that, given two bodies of water identical in these parameters, and differing only in initial uniform temperatures, the hot one will freeze sooner&#x201D;, it is invalid for the following reason: the initial parameters aren't correctly equivalents, because all conditions aren&#x2019;t the same at any time. Similarly, it's obvious that: &#x201C;hot water freezes faster than cold&#x201D;, particularly if:</p>
<p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn17.png"/></p>
<p>and/or:</p>
<p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn18.png"/></p>
<p>Therefore, only if these different parameters/conditions are satisfied, then there exists a pair of temperatures, such that for the two water samples: the hottest water one will freeze sooner.</p>
</sec>
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