Unification of the Fundamental Equations in Physics: A Single Functional Relation Between Mass and Volume

Indian Journal of Advanced Physics (IJAP) ISSN: 2582-8983 (Online), Volume-5 Issue-2, October 2025 14 Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Retrieval Number:100.1/ijap.B106605021025 DOI:10.54105/ijap.B1066.05021025 Journal Website: www.ijap.latticescipub.com Unification of the Fundamental Equations in Physics: A Single Functional Relation Between Mass and Volume Nishant Sahdev, Chinmoy Bhattacharya Abstract: Since the inception of classical physics, scientists worldwide have continually introduced new equations. Notably, the three fundamental physical variables—mass, time, and temperature—have consistently played central roles in these formulations. However, these variables are often treated as ‘abstract’ quantities, particularly because their physical dimensions, about length (L), have seldom been explicitly addressed. In this research, a novel thermodynamic approach has been employed to investigate the dimensional relationships between time (t), mass (M), and temperature (T) and length (L). The new dimensions revealed for these variables have the potential to reshape both modern and classical physics. The equation mass = density × volume holds profound significance in physics. Density, in this context, serves as an indirect measure of a material’s cohesiveness, or orderliness; among materials of varying density, the one with the greater density will house more molecules within a fixed volume, and is thus more cohesive. Density is, therefore, an index of order. Conversely, increasing the volume of a given mass—such as by introducing air voids or foams—decreases density, thereby diminishing cohesiveness and increasing disorder or randomness. Density, then, is the “creator” of order, while volume gives rise to disorder or randomness. Thus, mass, being the product of density and volume, is inherently a composite variable that embodies both order and disorder. This article demonstrates that the contribution of density (order) outweighs that of volume (randomness), making mass, as a hybrid parameter, primarily indicative of order. Every physical, mechanical, or chemical process is fundamentally an interplay of order and disorder, manifested in the equation mass = density × volume. Core principles of physics—including the laws of motion, mass-energy equivalence, ideal gas behaviour, wave-particle duality, the uncertainty principle, and the quantum mechanics of microscopic particles—can all be understood as manifestations of these order–disorder phenomena. Consequently, all fundamental equations in physics may ultimately converge upon, or be unified with, the relation mass = density × volume. Such unification or convergence can only be fully understood once the three principal variables—mass, time, and temperature—lose their abstraction and are properly embodied. In this research, we offer a foundational embodiment for these variables, opening new horizons for modern physics. Manuscript received on 26 July 2025 | First Revised Manuscript received on 08 August 2025 | Second Revised Manuscript received on 27 September 2025 | Manuscript Accepted on 15 October 2025 | Manuscript published on 30 October 2025. Correspondence Author(s) Nishant Sahdev, Department of Physics, University of North Carolina at Chapel Hill, US, India. Email ID: [email protected], ORCID ID: 0009-0007-2249-1006 Chinmoy Bhattacharya*, Austin Paints & Chemicals Private Limited, 3 Ambika Mukherjee Road, Belghoria, Kolkata (West Bengal), India. Email ID: [email protected], ORCID ID: 0000-0002-1962-0758 © The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Keywords: Dimensional Analysis, Mass-Volume Relation, Unified Physical Equations, Thermodynamic Compactness & Hard-Core Volume (HCV). Abbreviations: HCV: Hard-Core Volume FV: Free Volume KE: Kinetic Energies PE: Potential Energies I. INTRODUCTION The Pressure (P) parameter (as claimed to be an intensive property) of an ideal gas is expressed as ‘energy per unit volume’ [1] (P = nRT/V, R is the universal gas constant, T and V are the absolute temperature and volume, respectively, of the gas). As per the ideal gas equation, if at constant temperature and volume, more molecules (increasing the number of moles, n) are being introduced in a system of gas, the pressure does increase. Since the number of moles increases, the mass of the system also increases. Consequently, pressure becomes directly proportional to the mass, as (RT/V) remains a constant. The label of intensive character is being imposed on the pressure parameter, but it does not stand. It is argued that since the pressure of any solid substance or liquid (any mass of matter) remains constant when kept at ambient temperature and left open, the atmosphere exhibits the same pressure, which is atmospheric pressure; hence, it is intrinsic. However, this argument is flawed, despite its widespread dissemination throughout the global scientific community. The model of pressure has been developed (in the kinetic theory of gas [1]) for a constant temperature and volume condition, and pressure is shown as energy per unit volume. However, when it is stated that a sample of x grams of a substance has the same pressure as another n-gram sample (where n is a positive integer) of the same substance, it contradicts the model of pressure mentioned above. To follow the model, both the samples of x gm and nx gm must be brought under the exact condition of volume and temperature. If n > 1, the sample of higher mass must be compressed to achieve the same volume as the sample of x grams, and in that case, the pressure will be higher for the sample of nx grams than for the sample of x grams. If n<1, the reverse would be true. Hence, according to the kinetic theory of gases, pressure is highly dependent on the mass of the gas. So, pressure cannot be called an intrinsic property based on this model. Therefore, the definition of pressure in terms of energy per unit volume is falsifiable, as it cannot retain the ‘intensive physical variable’ characteristic of the Unification of the Fundamental Equations in Physics: A Single Functional Relation Between Mass and Volume 15 Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Retrieval Number:100.1/ijap.B106605021025 DOI:10.54105/ijap.B1066.05021025 Journal Website: www.ijap.latticescipub.com pressure parameter. Therefore, a new model of pressure needs to be developed, one that retains its intensive characteristic. When one takes an attempt to insert in a substance (of a certain specific mass) more molecules of the same substance (be it a liquid or solid or gas at a pressure P) without altering the temperature and volume of the substance, the pressure has to be exerted externally by an amount (P +∆P, ∆P is positive) to do this task. So, the said pressure (P +∆P) becomes the new pressure of the substance. After increasing the mass, the pressure increases; therefore, it cannot be an intensive property, as per the model of the kinetic theory of gases. The critical physical parameter in Physics is ‘pressure’, which has been assigned a dimension of ‘energy per unit of volume’. The problem in science is that the absolute definition of pressure could not be developed or proposed by anyone till the time of proposing the kinetic theory of gas, but that too was a ‘relative’ definition and not an ‘absolute’ one since it imposed the condition of constancy of both the ‘temperature’ and ‘volume’. The kinetic theory of gas [1] [2] took the help of the principle of Newton’s second law of motion (which states force is directly proportional to the rate of change of momentum). This law is purely empirical, and Newton did not provide any supporting evidence (regarding physics, thermodynamics, and topology) for the principle in question. The definition of pressure was being developed in the following manner: i) In a closed vessel containing gas molecules, the force exerted by the molecules on the wall of the vessel is equated to ‘change in momentum of the molecules after and before striking the wall per unit time’(t): (which is the Newton’s second law, however, which says force is directly proportional to the rate of change of momentum, not being equal and how far logical is it to apply his law for a closed thermodynamic system since the law was proposed for the mechanical motions of objects itself in open systems). =[𝑚𝑣−𝑚𝑢 𝑡] … (1) [m, u and v stand for mass, initial velocity and final velocity of the molecules, respectively]. ii) Then pressure is defined as force per unit area (A) =[𝑚𝑣−𝑚𝑢 𝑡] [𝐴] … (2) Suppose the above model of pressure is being examined analytically. In that case, it turns out that for the mechanical motions of the objects (when the object itself is changing its position as a function of time) the linking of pressure/force with the ‘change in momentum’ (as Newton did in his second law of motion) is not much logical since Newton did not thermodynamically explain the hypothesis, nor did he give any experimental proof of the same. Pressure is a parameter which is more related to volume rather than the ‘velocity ‘as in the above equation (1), hence the ‘temperature’ parameter has to be considered since the volume is dependent on temperature. One of the defects of this model of pressure calculation is that temperature has not been considered, which is a matter of ignoring the ‘thermodynamics of motion’. Now, according to the kinetic theory of gas, the expression for the average velocity of the molecules of a gas is, 𝑣𝑎𝑣=√(8𝑅𝑇 𝜋𝑀)=√(8𝑅𝑇 𝜋𝑁0𝑚) … (3) [R = universal gas constant, T = temperature in Kelvin, M is the molecular weight of the gas, and N0 is the Avogadro number]. If the dimension of (mv/tA) is being considered regarding both ‘time’ and ‘temperature’, its dimension comes out to be as under, which would be the dimension of pressure as per equations (1) and (2) above. (mv tA)=𝑚[√(8𝑅𝑇 𝜋𝑁0𝑚)] [𝑡𝐴]=𝑚√[𝑘√(𝑅𝑇 𝑚)] [𝑡𝐴] [𝑘=√(8 𝜋𝑁0)] =√[√(𝑘𝑅𝑇 𝑡𝐴)] … (4) So, it turns out from equation (4), when the ‘mechanics’ and the ‘thermodynamics’ of motion are both being considered, the ‘pressure’ becomes a function of both the ‘time’ and ‘temperature’ and not only the time as is done in conventional Physics. Later on in this article, equation (4) will be split further into dimension [1]. The dimension of pressure in the form of ‘energy per unit volume’ does not fit the model of Boyle’s law either. The said law is, when the temperature (T) remains constant, the pressure (P) is inversely proportional to the volume (V) of a gas (number of moles n is continuous), and it is mathematically expressed as, P∝1 V or P=(k V )[k is Boyle’s constant] … (4a) Since 𝑃= (energy volume )= (E V ), So, equation (4a) takes the form, (E V )= (k V ) … (4𝑏) In equation (4b) above, the LHS and the RHS contain the same variable V. However, in a proportionality relation, the LHS and the RHS cannot contain the same variable. So, the dimension of pressure in the form of ‘energy per unit volume’ is not acceptable. In the context of this article, especially to grasp the new concept, new logic, and new philosophy regarding the ‘pressure’ parameter, the understanding of the physical significance of the Boyle’s constant (k) is fundamental. When a gas in a closed system is compressed at a Indian Journal of Advanced Physics (IJAP) ISSN: 2582-8983 (Online), Volume-5 Issue-2, October 2025 16 Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Retrieval Number:100.1/ijap.B106605021025 DOI:10.54105/ijap.B1066.05021025 Journal Website: www.ijap.latticescipub.com constant temperature, keeping the number of moles of the gas constant, both the volume and the free volume (FV) decrease, but the total hard-core volume (HCV) of the gas molecules remains constant. While a gas is being expanded (at constant temperature, keeping the number of moles of gas, n, constant), both the volume and free volume increase; however, the total hard-core volume of the gas molecules remains constant. Regarding Boyle’s law, it can be concluded logically that Boyle's constant k is HCV per mole of the gas. The perfect model of pressure should be based on volume since volume is connected to HCV, FV, the mass of the molecules (m), time (t), temperature (T) and area (A) [the physical variables in equation (4)] as explained below: i) Mass Dependency: As the mass of the molecules increases, the density of the gas increases and the volume per unit of mass, or the specific volume, changes. ii) Time Dependency: For thermal expansion/contraction of a gas, the volume decreases or increases as a function of time. iii) Temperature Dependency: The volume of a gas is very much dependent on temperature. iv) Area Dependency: Volume itself is a function of area and length. Volume = (area x length) v) Dependency on HCV and FV: Volume of a gas, V = (HCV + FV) The volume of a substance, irrespective of whether it is a gas, liquid or solid, is the sum of two volumes: Total Volume = [Free Volume (FV) + Total Hard-Core Volume of the molecules (HCV)] As shown in Figure 1 below, as the piston of the cylinder containing a gas is being put more downwards, the pressure (P) increases, volume (V) decreases, and the temperature (T) increases (as follows from the ideal gas equation PV = RT). At any stage of the said compression, as shown in Figure 1, the HCV (total) remains constant, but what changes is the volume (V) of the gas. The higher the magnitude of force on the piston, the lower the ratio of (HCV/V) becomes, since V is decreasing and HCV remains constant. From one gas to another gas, though the mass of the molecules would be varying, the force on the piston will remain proportional to the ratio of HCV to V. When the said ratio is higher, the force on the piston will be lower, and when the said ratio is lower, the force will be higher. So, 𝐹𝑜𝑟𝑐𝑒 𝑜𝑛 𝑡ℎ𝑒 𝑝𝑖𝑠𝑡𝑜𝑛 𝛼 [𝐻𝐶𝑉 𝑉] … (5) [Fig.1: Representation of Pressure as a Function of Volume (V) and Hard-Core Volume (HCV)] The relation as shown in equation (5) is a universal one and will not at all be dependent on how the process is being carried out (isothermal, adiabatic or constant pressure condition), and the said proportionality of equation (5) or this suggested model of pressure is applicable for any substance, whether it be a solid, liquid or gas. Any generalised expression of ‘force’ must be valid for all three states of substances, i.e., gas, liquid, and solid. Additionally, the mathematical expression of force (unlike the equation derived in the kinetic theory of gases) should also be applicable for varying temperatures and volumes. The subjects of Physics and physics are being overly burdened with the problems of circularity in the definitions of physical variables. For example, Force is depicted as (pressure × area), and on the other hand, pressure is defined as force per unit area. Mass is being defined as (density x volume), and density is being defined as (mass/volume). However, no fundamental definition of mass has been created. Velocity is defined as Distance per unit of time, and time is defined as (distance/velocity). However, ‘time’ is not being defined fundamentally and has been labelled as an ‘abstract’ variable. However, many such examples would be found in this cycle. From Figure 1, it can be observed that as the piston is moved downwards, the temperature of the system increases. Conversely, when the piston is pushed upwards from a downward position, the temperature decreases. Hence, this is a case of varying temperature and volume. When the piston is moved downwards, the volume decreases; there is a change in the length (height) of the piston, and the reverse is true. So, the Force can be defined fundamentally from the Unification of the Fundamental Equations in Physics: A Single Functional Relation Between Mass and Volume 17 Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Retrieval Number:100.1/ijap.B106605021025 DOI:10.54105/ijap.B1066.05021025 Journal Website: www.ijap.latticescipub.com thermodynamic point of view as being proportional to the rate of change of volume concerning the height or depth of the substance (vis–à–vis length) as shown below, Force ∝ (Rate of change of volume with depth or length) 𝑘(𝑑𝑉 𝑑𝐿)= 𝐿0(𝐿3 𝐿)=𝐿2 Or,Force =𝑘 (𝑑𝑉 𝑑𝐿)= 𝐿0(𝐿3 𝐿)=𝐿2 … (5𝑎) The proportionality constant, k in equation (5a), is the ‘ratio of the total hard-core volume of the molecules (HCV) to the total free volume (FV) of the material’. It will differ from one material to another and will be dimensionless since it is the ratio of two volumes. For the gases, the (THCV) << (FV), the magnitude of k would be lower. However, for liquids, the said ratio (HCV/FV) is typically higher (for water, it is approximately 63%), and the k value is higher for liquids than for gases. For solids, the ratio is typically higher (for naphthalene, it is approximately 170%). So, the k value, in general, would be the highest for the solids. Since in the case of gases, the free volumes are relatively high, and as a result, upon application of a small force, the parameter (dV/dL) will attain a high value. The higher the (dV/dL), the higher the compressibility of a substance. In contrast to that, the magnitude of (dV/dL) would be very low for solids since the HCVs are higher. While a gaseous substance, upon application of a force of amount F, will be compressed by a percentage of x%, a solid substance, upon application of the same force, F, will be compressed by a percentage of << x%. Therefore, the magnitude of (dV/dL) for the gaseous substance and the solid substance would differ significantly. The parameter k in equation (5a) (being the ratio of HCV to FV) represents the ‘compactness’ of a substance. The higher the ratio of (HCV/FV), the more compact a substance would be, and the reverse would be true. While k represents the ‘degree of compactness of a substance’, the parameter in the form of the differential coefficient of V and depth (length), (dV/dL), represents the ‘degree of deformation’ of a substance. The hybrid effect of ‘degree of compactness’(k) and the ‘degree of deformation' (dV/dL) of a substance is indeed what the force is. The ’degree of compactness’, k, does take into account the ‘intermolecular attractive forces’ of a substance. For any substance in which the (HCV to FV) ratio is high, it originates from the higher level of cohesiveness among the molecules, causing them to come closer and closer to each other, which reduces the free volume. As the free volume decreases, the HCV-to-FV ratio increases to a higher value. So, the model of Force or pressure as is being presented in this article is an universal model and should not be considered as a model for the ideal gases only (or for the perfect behavior of the molecules of a substance) as had been done in the model of kinetic theory of gas where the effect of intermolecular attractive forces among the molecules were being entirely ignored. The calculation of (HCV/FV) for oxygen gas, water, and naphthalene at 298K, based on their literature-reported molecular diameters [3] and empirical data obtained, is shown in Table 1 below. [The principle of calculation: 1 mole of a substance contains the same number of molecules, which is the Avogadro number, N (= 6.023 x 1023). Considering the molecules to be spherical with radius (r), the mathematical expressions of Volume (V), HCV and FV would be, 𝐻𝐶𝑉=𝑁[4𝜋𝑟3 3] V = [molecular weight /density], density obtained from literature. FV=[V−N(4𝜋𝑟3 3)] However, the magnitude of the parameter (dV/dL) will be dependent on the geometry of the object or the substance. The typical plots of V versus L for the compression of a gas in a cubic cylinder and the compression of a spherical substance (uniformly compressed, despite its surface area) are shown in Figures 2 and 3 below. The empirical data for the figures are presented in Table 2a and Table 2b, respectively. The calculation of (dV/dL) has been made for two cases: i) a gaseous substance in a cubical cylinder with a piston, and ii) a solid spherical substance (not hollow). For the cubical cylinder (of length L), as the piston is moved downwards, the length and breadth remain at L each, and the height (or depth) does change. If the depth is xi at any stage, the volume of the gas would be (L² xi). The plot has been created of volume versus xi at different stages of compression. Table 2a: Empirical Data of HCV and FV of substances Substance Radius (cm) Molar Volume (cm³/mol) Va Hard Core Volume, HCV (cm³/mol) N [4πr3/3] Free Volume (cm³/mol), FV [V - N (4πr3/3)] HC Volume/ Volume (HCV/V) HC Volume/ Free Volume (HCV/FV) Oxygen (O₂) 1.52 × 10⁻⁸ 22,400 8.86 22,391.14 0.0004 0.0004 Water (H₂O) 1.40 × 10⁻⁸ 18.07 6.92 11.15 0.383 0.6209 Naphthalene 3.00 × 10⁻⁸ 108 68.11 39.89 0.6306 1.7073 Table 3a: Empirical Data of Depth vis-à-vis Volumes of the Gases at the Different Stages of Compression Depth (xi) Percent Compression (%) V (yi) 10.00 0% 1000.00 5.00 50% 500.00 2.50 25% 250.00 1.25 12.5% 125.00 0.625 6.25% 62.50 0.3125 3.125% 31.25 Indian Journal of Advanced Physics (IJAP) ISSN: 2582-8983 (Online), Volume-5 Issue-2, October 2025 18 Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Retrieval Number:100.1/ijap.B106605021025 DOI:10.54105/ijap.B1066.05021025 Journal Website: www.ijap.latticescipub.com [Fig.2: Plot of Volume Versus Depth at Different Stages of Compression of a Gaseous Substance (Data Table 3a)] Table 2b: Empirical Data of Compression (Radius and Volume) of a Spherical Solid Substance, although its External Surface Area is Uniformly S. No. Volume Expression Volume (Approx) Volume Compression (%) Radius Expression Radius (Approx) 1 V1 = 4000 𝜋/3 4188.79 0% R1 = (3V1/4𝜋)1/3 10.00 2 V2 = V1/2 2094.40 50% R2 = (3V2/4𝜋)1/3 7.93 3 V3 = V1/4 1047.20 25% R3 = (3V3/4𝜋)1/3 6.29 4 V4 = V1/8 523.60 12.5% R4 = (3V4/4𝜋)1/3 5.00 5 V5 = V1/16 261.80 6.25% R5 = (3V5/4𝜋)1/3 3.96 6 V6 = V1/32 130.90 3.125% R6 = (3V6/4𝜋)1/3 3.14 7 V7 = V1/64 65.45 1.5625% R7 = (3V7/4𝜋)1/3 2.50 [Fig.3: Plot of Volume Versus Radius of a Solid at the Different Stages of Compression (Data from Table 3b)] The expression of pressure as obtained from equation (5a) is, Pressure=P=(force per unit area)= (𝐿2 𝐿2)=𝐿0=dimensionless … (5b) Now, as per the model of force/pressure in this article, Force = (Compactness of the substance, k) x (degree of deformation, (𝑑𝑉 𝑑𝐿) … (5c) Now, (dV/dL) has the dimension of area, (=𝐿ˆ3 𝐿)=𝐿ˆ2. So, Pressure = (Force /area) = k (compactness of the substance) (5d) Since compactness (= HCV/FV) does not depend on the mass of the substance, the pressure parameter turns out to be an ‘intensive’ property. The compactness of a substance is a function of temperature only, like the ‘refractive index’ of a substance. The refractive index is an intensive property that also depends on temperature. So not only does equation (5) reveal that parameter P is a dimensionless parameter (since it is a ratio of two volumes only), but it is also found from equations (5a) and (5b) too. Newton, while proposing that ‘force is directly proportional to the rate of change of momentum’, had tactfully avoided the proportionality constant and equated force straightaway to (mass x acceleration), which is not acceptable. However, as shown in equation (5a), in this article, the proportionality constant has been retained in the newly proposed equation of force, and the proportionality constant has to be there. The mathematical expression of force and pressure [Equation (5a), (5b) and (5c)] are the generalized and absolute expressions of the said variables without any constraint (like constant temperature and constant volume) being imposed on them and is being based on the dimension of L only, since any volume be it V, HCV or FV holds a dimension L3. Since only through dimension L have the mathematical expressions of ‘force’ and ‘pressure’ been obtained in this article, and for this apparent reason, it is called ‘absolute’. The physical variable is defined as ‘force’ is not a standard term in physics. However, it is introduced in Unification of the Fundamental Equations in Physics: A Single Functional Relation Between Mass and Volume 19 Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Retrieval Number:100.1/ijap.B106605021025 DOI:10.54105/ijap.B1066.05021025 Journal Website: www.ijap.latticescipub.com this research article in a tripartite manner, about mathematics, Physics, and topology or geometry. As per Physics, the dimensions of ‘energy’ and ‘volume’ are: 𝐸𝑛𝑒𝑟𝑔𝑦 =𝑀 𝐿2 𝑇2=𝑃𝑉 … (6) 𝑉𝑜𝑙𝑢𝑚𝑒 =𝐿3 =𝑉 … (7) Since P is dimensionless, as is being proved, the dimension of ‘energy’ and ‘volume’ merges to the same point, and that is being L3 (or volume) and hence the dimensions of energy and volume are the same indeed. Comparing the equations (6) and (7) it (since P is dimensionless), one gets, 𝑀𝑇−2 = 𝐿 … (8) Therefore, the derived equation (8) is significant in physics, as it shows that the three dimensions, L, M & T, are interconnected with each other. So, it is a diametrically opposite finding to that of conventional Physics, where the three principal dimensions L, M & T are considered to be independent of each other. Equation (8) has far-reaching implications, which will become apparent to the readers as this article progresses further in the subsequent sections. A. Law of Conservation of Mass-Energy and the Dimensions of ‘Mass’ and ‘Density’ The equation for energy is: Energy = Force x distance … (9) Now equation (9) can be rearranged in its reciprocal form, 1 𝑒𝑛𝑒𝑟𝑔𝑦=1 𝑓𝑜𝑟𝑐𝑒 x 1 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒… (10) Now, for the time being, setting aside equations (9) and (10), let us examine how the law of conservation of energymass (E & M) in science can be mathematically presented. The law states that ‘the total amount of mass and energy remains constant and does not change.’ The possible mathematical statement can be multiple, as under: [𝐸 + 𝑀]= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 … (11) [𝐸/𝑀] = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 … (12) [𝐸 𝑥 𝑀]= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 … (13) However, equation (11) is not acceptable since ‘energy’ and ‘mass’ have different dimensions and cannot be added up to each other. Equation (12) is also not acceptable since, as the mass tends to zero, the energy would also tend to zero. If both the mass and energy became zero, the universe would not have existed at all. Therefore, the only equation that would be acceptable is equation (13), as in this equation neither the mass nor the energy is allowed to attain values of ‘zero’ or ‘infinity’. Equation (13) is the hybrid form of ‘energy’ and ‘mass’. Now the dimensionality of equation (13) would be: (𝐸 𝓍 𝑀)=( 𝑀𝐿2𝑇−2 𝑥 𝑀)=(𝑀𝐿𝑇−1)2 =(𝑖𝑚𝑝𝑢𝑙𝑠𝑒)2 … (14) The dimension of ‘impulse’ is M L T-1, and impulse has been defined as the change in momentum of an object when a force acts upon the object for some time ∆t, and is the change in momentum of an object arising from a sudden collision with another object. Since it has been proved in this article that ‘pressure’ is a dimensionless parameter, impulse is dimensionless too. Therefore, the product of energy and mass is a constant quantity, as shown in Equation (13), and is dimensionless. Now returning to equation (10), it can be written in the following form based on equation (13), [( 𝑀 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝑙𝑒𝑠𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝑘))] =[ 1 (𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑥 𝑎𝑟𝑒𝑎)]𝑥[ 1 𝑑𝑖𝑡𝑎𝑛𝑐𝑒] =[𝑘1(1 𝑎𝑟𝑒𝑎)𝑥( 1 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒)] … (15) [Since P is dimensionless and (1/P) = constant at a specified temperature = k1] Now, leaving the dimensionless constants both in the LHS and RHS and retaining only the dimensional or variable parts of equation (15), it can be written as: 𝑀=(1𝑉)[𝑠𝑖𝑛𝑐𝑒 (𝑎𝑟𝑒𝑎 𝑥 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒=𝑣𝑜𝑙𝑢𝑚𝑒)] =(1 𝐿3)(1/𝐿3) … (15𝑎) 𝑀 =(1/𝑉2𝑥 (𝑉) … (16) Now, if equation (16) is being compared with the following most fundamental equation in Physics, 𝑀𝑎𝑠𝑠 (𝑀)=(𝑑𝑒𝑛𝑠𝑖𝑡𝑦)𝑥 (𝑣𝑜𝑙𝑢𝑚𝑒)=(𝜌 𝑉) … (17) The dimension of density (ρ) is obtained to be, 𝐷𝑒𝑛𝑠𝑖𝑡𝑦= 𝜌=(1 𝑉2) … (18) Or,ρ=(1 𝑉6)[since V=L3] … (19) B. Dimension of ‘Time’ and ‘Temperature’ From equation (8), it has been found that, 𝑀𝑇−2=𝐿 Now, in equation (16), it has been established that M = (1/V), and so putting it in equation (8), one gets, (1𝑉)𝑇−2=𝐿 Now since 𝑉=𝐿3,𝑇2=(1 𝐿3)𝑥(1𝐿)=(1 𝐿4) Or,Time=t=(1 𝑙2) … (20) [Time from now will be presented in this article as t instead of T of the conventional physics to Indian Journal of Advanced Physics (IJAP) ISSN: 2582-8983 (Online), Volume-5 Issue-2, October 2025 20 Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Retrieval Number:100.1/ijap.B106605021025 DOI:10.54105/ijap.B1066.05021025 Journal Website: www.ijap.latticescipub.com avoid the mix-up with temperature, which will be represented by T in this article]. Now in thermodynamics, the Boltzmann constant k = (Energy Temperature)= (ET)… (21) The mathematical expression for energy in thermodynamics is E kT, and this can be written as, 𝐸 = ( 𝐸𝑛𝑒𝑟𝑔𝑦 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒)𝑥 (𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒) = (𝐸𝑇)𝑥 (𝑇) … (22) The Planck’s expression for energy is, 𝐸 = ℎ√= (𝐸𝑛𝑒𝑟𝑔𝑦 𝑥 𝑇𝑖𝑚𝑒)𝑥 ( 1 𝑇𝑖𝑚𝑒) = (𝐸𝑡)𝑥 (1𝑡) … (23) [h = Planck’s constant = Energy. sec and √ = frequency = (1/second)] Now, if the two equations (22) and (23) are compared to each other, one finds, 𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 = 𝑇 =(1𝑡) … (24) 𝑂𝑟,𝑇𝑡 = 1 … (25) So, from equation (24), it is being established that ‘temperature’ and ‘time’ are multiplicative inverses to each other [1], and it is also being found that, 𝑇=𝐿2 … (26) The dimension of the universal gas constant/Boltzmann constant can now be evaluated from the ideal gas equation, PV = RT, as shown below: From the ideal gas equation, it follows that R = (PV/T) = (volume/temperature) [since P is dimensionless]. = (𝐿3 𝐿2)=𝐿 … (27) The Boltzmann constant = (R/N0) [N0 is Avogadro’s number and is dimensionless]. So, both R and k have the same dimension, which is distance, L. The expression for velocity is (L/t), and the concept of ‘velocity’ merges with the concept of ‘volume’. since t = (1/L2), and hence, velocity becomes [(L/(1/L2)] = L3 = volume = V. The physical concept behind this phenomenon is that, for an object to move, it has to overcome the ‘time frame’ (t), which does exist in the inverse form of (length)2 (is an attractive force). So, once it overcomes the time-attractive force, it converges to a volume. Velocity is related to volume in the sense that when the velocity of the molecules of a gas increases upon heating at constant pressure, the volume also increases. 1. Dimension of momentum (p), Planck’s constant (h), gravitational constant of Newton (G), acceleration due to gravity (g)/acceleration (f), kinetic energy (E), potential energy (V), stefanboltzmann constant (σ), surface tension (ϒ), viscositycoefficient(ղ), electric charge (Q), current (I), voltage(V), electrical resistance (R) and wave function ψ of quantum mechanics. [Fig.3: Physics of Formation, Mathematical Expression and the Geometrical Shape] [Fig.4: The Concept of Integrated and Differential Forms of Physical Variables] Regarding the dimensions of the physical variables, the following points are to be noted: i) Every physical variable has to have a geometrical shape: there must exist a function of its evolution, and finally, for the measurement of its magnitude, a mathematical formula has to be in place. This is referred to as a ‘tripartite’ representation of a physical variable, as illustrated in Figure 3 above. ii) Any physical variable has three forms: the integrated form, the partial differentiation/partial integrated form and the fully differential form. iii) When a person is observing an object, such as a 3D round shape object, it might so happen that two 3D spheres are coexisting in one round object. This is the integrated form, where the dimensionality is six inherently. But when the same object is being shown in its differential form Unification of the Fundamental Equations in Physics: A Single Functional Relation Between Mass and Volume 21 Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Retrieval Number:100.1/ijap.B106605021025 DOI:10.54105/ijap.B1066.05021025 Journal Website: www.ijap.latticescipub.com (separating the two spheres from each other and being placed side by side), what would be observable to a person is that two 3D objects are lying adjacent to each other. The concept of ‘integrated form’, ‘partial integrated/differential form’ and the ‘complete differential form of a physical variable is being shown pictorially in Figure 4 above. iv) The entire concept of ‘dimensionality’ of the physical variables has to be rebuilt. As it will be shown later in this article, the dimensionality of the ‘acceleration’ parameter is 5. The physical significance of the same is that it is composed of a 3D sphere and a 2D circle. One would find it in a 3D form of shape (which represents the integrated state), but in its differential form, it would be found as a combination of a 3D sphere and a 2D circle. The Planck constant had been defined in Heisenberg’s uncertainty principle as: Et = h … (28) 𝑆𝑜, (𝐿3𝑥 1/𝐿2)=ℎ Or h = L … (29) The dimension of h turns out to be the ‘length’ only. Momentum, p, has been defined as, p = mu (m is mass and u the velocity of an object) So, 𝑝=(1 𝐿3)𝑥 (𝐿3)=1 So, momentum p is a dimensionless parameter. Now, entropy, S, in thermodynamics has been defined as (Q/T), where Q stands for energy, and T stands for temperature. So, Degree of Randomness = Entropy = S = (energy/temperature) =(𝐿3 𝐿2)=𝐿 … (30) Degree of order = (1/ Entropy) = (1/ S) = (temperature /energy) =(𝐿3 𝐿2) (1𝐿) … (30𝑎) The ‘acceleration(f) / acceleration due to gravity (g)= (𝐿 𝑇2)[(𝐿1 𝐿2)2]= 𝐿5 … (31) The dimensionality of the g/f reaches five dimensions. The Gravitational law of Newton is, 𝐹=𝐺(𝑚1𝑚2 𝑟2) … (32) The masses of the two objects are m1 and m2, respectively: r is the distance of separation; F is the force; and G is the gravitational constant. Rearranging equation (32), it can be written as, 𝐺=( 𝑟2 𝑚1𝑚2) … (33) Considering the masses, m1=m2 = m and putting the dimensions of force, distance and the masses in equation (33), 𝐺=[(𝐿2𝑥𝐿2)/(1/𝐿3)2𝐿10… (34) While the ‘acceleration’ is being L5, the gravitational constant attains the dimension L10. The phenomenon of gravitation arises out of the ‘overlapping’ of two acceleration fields, and the dimensionality in such a way reaches 10. Through this exercise, the dimensionality is also revealed. The kinetic energies (KE) and the potential energies (PE) are defined as, 𝐾𝐸= (𝑚𝑢2) 2 … (35) PE = mgh … (36) The parameter h in equation (36) represents height (length), and u is the velocity, which merges to form volume (L³). So, putting the dimensionalities of the physical variables in equations (35) and (36), one gets 𝐾𝐸=𝐸=(1 𝐿3𝑥𝐿6)=𝐿2 … (37) PE = V (as in quantum mechanics) =(1 𝐿3𝑥𝐿5𝑥𝐿)=𝐿3 … (38) The energy expression in the Stefan–Boltzmann law is, 𝐸=σ 𝑇2… (39) 𝑆𝑜,𝜎=(𝐸 𝑇4)=(𝐿3 (𝐿2))4]=(1 𝐿5 )… (40) The mathematical presentation of surface tension (ϒ) is, ϒ = (Force distance)= 𝐿2 𝐿=𝐿 … (41) The mathematical expression for the viscosity coefficient (ղ) is, (ղ)= [ force area velocity distance] … (42) Now putting the dimensions of the physical variables in equation (42), 𝑛⌈(𝐿2 𝐿2 𝐿3 𝐿)⌉ 𝑜𝑟 𝑛= (12 𝐿) … (43) So, the dimension of ‘viscosity coefficient’ and ‘time’ comes to be the same. Viscosity is the resistance to flow that arises from intermolecular attractive forces. As mentioned earlier in this article, ‘time’ is also an attractive force. Indian Journal of Advanced Physics (IJAP) ISSN: 2582-8983 (Online), Volume-5 Issue-2, October 2025 22 Published By: Lattice Science Publication (LSP) © Copyright: All rights reserved. Retrieval Number:100.1/ijap.B106605021025 DOI:10.54105/ijap.B1066.05021025 Journal Website: www.ijap.latticescipub.com The Columba’s law of the force (F) operating between the two arbitrary electric charges q1 and q2 at a distance r, is: 𝐹=𝑘(𝑞1𝑞2 𝑟2) … (44) [k is the Columba’s constant = (1/4π€0), where €0 stands for the permittivity and is the ratio of two capacitances, €0 = C/C0 and hence k is a dimensionless parameter] Putting the dimensionalities of the physical variables in equation (44), and with the assumption q1 = q2 = q, one gets, 𝑞2=( 𝑟2𝑥𝐹)=( 𝐿2𝑥 𝐿2)= 𝐿4 or electric charge 𝑞 = 𝐿2 … (45) The dimension of ‘current’ (I) is (T-1 Q), where T is time (1/L2) and Q is electric charge = q = L2 and hence, 𝐼=( 𝑇−1𝑄)=( 𝐿2𝑥 𝐿2)= 𝐿4 … (46) The dimension of ‘voltage’ is (L2MT-2 Q-1), where T stands for time and M for mass. So, the dimension of voltage (V) turns out to be, 𝑉=[( 𝐿21 𝐿3𝑥 𝐿4𝑥 𝐿−2)]=𝐿 … (47) Now the Ohm’s law is, Voltage = (Current x Resistance) = (I x R) … (48) Putting the just-now-derived dimensions of V and I, it is found, R = Resistance =(VI)=(L 𝐿4)=(1 𝐿3) … (49) So, the dimensions of ‘mass’ and ‘resistance’ converge on each other. Mass is an attractive force too (like time), and mass is a squeezing phenomenon that resists the current (I) from flowing. The angular momentum is defined as the product of mass, velocity, and radius. While m is the mass, v is the velocity, and r is the radius of an electron's orbit. So, the dimensionality of the angular momentum converges to: Angular momentum mvr=[(1 𝐿3)𝑥(𝐿3)𝑥 𝐿]=𝐿 … (50) Now the Schrodinger equation in quantum mechanics is, (𝑑2ψ 𝑑𝑥2)+(8𝜋2𝑚 ℎ2)[𝐸−𝑉]ψ=0 … (51) [Here ψ is the wave function, and h, E, m, and V stand for the Planck’s constant, mass, total energy and potential energies, respectively] Regarding equation (51), the following points are to be noted [1] [2]: i) The parameter ψ is in the form of a mechanical wave, and the second derivative of such a wave can attain both positive and negative values depending on its shape. On a point of the wave, if the said second derivative is positive, then the shape of the wave is convex or concave upwards. If the said derivative is negative at a point, the shape is concave or convex downwards. ii) The sum of the two terms in equation (51) has to be zero always. So, when the second derivative is positive (the first term), the second term must attain a negative value. In contrast, when the first term/second derivative is negative, the second term must achieve a positive value. iii) The value of the second term in equation (51) can attain positive or negative values depending on the following factors: (to note here that in the second term, none of the constant parameters h or m could be negative) a. When E > V (kinetic energy is higher than potential energy) – 2nd term is positive. b. When V >E (potential energy is higher than kinetic energy) – 2nd term is negative. iv) For the right-hand side of equation (51) to be zero, the following mathematical conditions have to be true: a. When (d2ψ/dx2) is positive, (E-V) is negative. b. When (d2ψ/dx2) is negative, (E-V) is positive. c. In either of the case modulus (d2ψ/dx2) = modulus (8π2m/h2) [ E -V] ψ The topology of ψ in equation (51) becomes a unique one, which could be solved only through an analytical approach through the model of a pair of adjacent circular orbitals in the space, as shown in Figure 4 below: [Fig.5: Topology Ψ Function of Quantum Mechanics (Integrated Form, Semi-Integrated Form and Differential Form)] In Figure 5(i), between the two orbitals A & B, orbital A is higher in size/area than orbital B. When A is stationary and B is rotating, then V is higher than E since the size of A is higher. When B is stationary, and A is moving, the reverse becomes true, i.e., E > V. [When an orbital is stationary, its entire energy is potential. As the orbital starts rotating, the potential energy begins to convert to kinetic energy. In this article, it has been established that the higher the volume, the higher the energy. So, the