Rubber As an Aid to Teach Thermodynamics∗ the Discovery by a Blind Scientist

GENERAL ARTICLE Rubber as an Aid to Teach Thermodynamics∗ The Discovery by a Blind Scientist

Geethamma V G and Sampath V

The behaviour of rubber diﬀers from that of conventional materials. Rubber heats up on stretching and cools on retraction. Also, stretched rubber shrinks on heating (thermoelastic shrinkage) while a stretched metal elongates. The elastic recovery of rubber is due to its tendency to maximize the entropy. The same property also causes the thermoelastic shrinkage. Metallic materials possess energy elastic- Geethamma V G was a ity, while ideal rubber possesses entropy elasticity. The ther- Fulbright Fellow at modynamic behaviour of rubber is similar to that of gaseous University of Illinois, USA materials. Hence rubber can be used as an aid for teaching and a Royal Society International Postdoctoral thermodynamics. Fellow at Cavendish Lab, University of Cambridge, UK. She was also a Young 1. Gough–Joule Eﬀect Scientist Awardee.

John Gough was not born blind. But by a sheer quirk of fate, he lost his eyesight due to smallpox before turning three. However, his senses of touch and hearing were intact, and he was bestowed with a sharp inquisitive mind and adequate skills for experimen- V Sampath is presently a tation. He tended to be philosophical too. John Dalton, a well Professor of Metallurgical known British scientist, who proposed the atomic theory, helped and Materials Engg. at IIT Gough by reading out to and writing for him. He had great admi- Madras. He holds a PhD from IISc, Bengaluru and has ration for Gough. three decades of research and In 1805, Gough observed two important properties of rubber [1]. teaching experience. His ﬁelds of interest are: shape Firstly, in a simple experiment, he sensed with his lips (lips being memory alloys, smart a temperature sensitive part of the human body), a rise in temper- materials, nano- and ature when a piece of uncrosslinked natural rubber was stretched composite materials, and rapidly. He also discovered that the rubber cooled down rapidly structure-property correlation in materials. on retraction. In another experiment, a rubber band (suspended

∗DOI: https://doi.org/10.1007/s12045-019-0772-x

RESONANCE | February 2019 217 GENERAL ARTICLE

Box 1. Gough–Joule Eﬀect

1. Rubber warms on stretching and cools down on retraction. 2. Stretched rubber shrinks on heating and elongates on cooling.

Experience Gough–Joule Eﬀect! Take a piece of rubber with the dimensions 7 × 3 cm. Hold it loosely on the upper lip. Stretch it quickly to the maximum extent possible, taking care not to break it. Your lip will feel the warmth. Then hold the strip away from the lip for 5 seconds. Place it on the lip and release the force quickly; now your lip becomes cold.

Keywords with a weight) was warmed. It was observed that the rubber band Gough–Joule effect, rubber, ther- contracted thus raising the weight. Instead, if the length of the moelastic, entropy elasticity, adiabatic, isothermal, thermodynam- rubber band was kept constant, the tension in the rubber increased ics, teaching aid, strain-induced with temperature. So he observed that stretched rubber contracted crystallisation, elastomer. on heating and elongated on cooling. About fifty years later, another scientist named James Joule confirmed these observations with crosslinked natural rubber [2]. This behaviour is observed in certain synthetic rubbers too. This phenomenon is known as Gough–Joule Effect (Box 1) [3]. Since the phenomenon involves a temperature change when a solid is sub- jected to stress, it is also known as thermoelastic (elastocaloric) property [4]. However, if the stress exceeds the elastic limit, it is called thermoplasticity. In other words, the thermal response of an elastic material is thermoelasticity while the thermal response of plastic is thermoplasticity.

The phenomenon of Rapid deformation of metals also causes a change in tempera- shrinkage on heating and ture. But it is only in the range of one Kelvin. However, the expansion on cooling is phenomenon of shrinkage on heating and expansion on cooling is a unique property of stretched rubber. This a unique property of stretched rubber. This behaviour is in con- behaviour is in contrast trast to that observed in conventional materials which expand on to that observed in heating. conventional materials which expand on The increase in temperature during the deformation of rubber is heating. not fully oﬀset by the decrease in temperature during its retrac-

218 RESONANCE | February 2019 GENERAL ARTICLE tion. So Gough–Joule effect is significant in certain practical applications. Rubber products such as automobile tyres, bridge segments, and vibration dampers are exposed to cyclic loading during their service life. Heat is generated in these rubber products due to the conversion of mechanical energy into heat (hysteresis). Hysteretic heat build-up represents a wastage of mechanical energy. In addition, it leads to early damage to the product. The total heat developed in a product is the sum of hysteretic heat and heat due to the thermoelastic effect. In many products, heat generated due to thermoelasticity is crucial. A familiar example is the wear and tear of automobile tyre as it undergoes several deformation-recovery cycles during its rotation.

2. Thermoelastic Inversion

In Gough–Joule effect, the temperature of rubber increases con- The thermoelastic siderably when it is stretched to large extensions. But for small inversion occurs due to extensions (< 50%), a slight decrease in temperature is observed. the small but thermodynamically This phenomenon is called thermoelastic inversion [5]. However, significant changes of for small deformations, the temperature of rubber increases dur- volume occurring with a ing compression and does not even change during shear or tor- change of temperature or sion. Also, generally materials exhibit a positive coefficient of the application of a ffi force. This phenomenon linear expansion. But for rubber, this coe cient changes from is not shown by ideal positive to negative depending upon its extension. Stretched rub- rubbers where the ber shrinks on heating at fairly high extensions. At high elonga- change in volume is tion, the force developed in stretched rubber is proportional to the negligible. absolute temperature. But at low extensions (< 10%), stretched rubber elongates on heating. It indicates decreased tension at high temperature [6]. This phenomenon is also due to thermoelastic inversion. The thermoelastic inversion is due to the small but thermodynamically significant changes of volume occurring with a change of temperature or the application of a force. Hence this phenomenon is not shown by ideal rubbers where the change in volume is negligible. In brief, the competition between energy effects and entropy effects in rubber at small and large extensions leads to thermoelastic inversion.

RESONANCE | February 2019 219 GENERAL ARTICLE

Figure 1. Isothermal and adiabatic processes.

3. The Secret Behind Gough–Joule Eﬀect

In general, an object feels cold when heat flows from our skin to the object. Conversely, it feels warm when heat flows from the object to our skin. The stretched rubber strip feels warm because it liberates heat to our skin. Now let us see, how heat is developed in rubber during stretching. The reason of Gough–Joule effect remained elusive for many years until Herman Staudinger, Kuhn and others propounded theories in the 1920s and 1930s [7–9]. The two fundamental requirements for a material to exhibit thermoelasticity are temperature change and elasticity (reversibility). At first, consider the temperature effect. There are two reasons for this: adiabatic stretching and strain-induced crystallisation.

In an isothermal process, The temperature of a system increases or decreases depending the system is in contact upon the nature of phenomenon occurring in the system. Let us with a reservoir at a see the diﬀerence between isothermal and adiabatic processes. In constant temperature, while in an adiabatic an isothermal process, the system is in contact with a reservoir process, the system is at a constant temperature. Thermodynamically speaking, a reser- thermally insulated from voir is a large entity which can transfer heat into or out of a system its surroundings. without undergoing a change in temperature. Hence in an isothermal process, the system either receives heat or loses heat so that its temperature remains constant (Figure 1). The transformation

220 RESONANCE | February 2019 GENERAL ARTICLE is very slow in an isothermal process. But in an adiabatic process, the system is thermally insulated from its surroundings. The insulator can be rubber, plastic or wood. These materials have very low thermal conductivity and the system neither gains heat nor loses heat from or to the surroundings. Hence though the system undergoes a process which may increase (or decrease) its temperature, heat is not emitted (or absorbed). In this process, energy is exchanged with the surroundings only as work. When the process is fast, it does not have enough time for the transfer of energy as heat to or from the system. So when a material is deformed rapidly, the temperature of the system is changed. The experiment mentioned in Box 1 should be rapid so as to minimize heat transfer from the rubber strip to the surroundings, and thus make the process adiabatic.

Normally rubber is amorphous. But certain types of rubbers crys- Normally rubber is tallise on fast stretching due to strain-induced crystallisation. This amorphous. But certain property is discussed in detail later. Simultaneously, the temper- types of rubbers latent heat of crystalli- crystallise on fast ature of rubber rises due to the release of stretching due to sation. This explains why the lip feels warm as discussed earlier. strain-induced The latent heat of crystallisation raises the temperature of rubber crystallisation. up to 10 K at 500% strain. But the adiabatic loss of its entropy Simultaneously, the ﬀ temperature of rubber can reach only 1 K at this strain. Hence Gough–Joule e ect is rises due to the release of mainly due to the latent heat of crystallisation. When the force latent heat of is released, rubber retracts to its original state becoming amor- crystallisation. phous again absorbing heat from the surroundings. This leads to the sensation of lips getting cooled. The second observation, viz., shrinkage-on-heating, can be ex- plained as follows. If heat is released during the stretching of rubber, adding heat causes its contraction. The enhanced thermal energy of rubber molecules increases the possibility of conformational changes. This accelerates the recovery of rubber. Also, the elastic modulus of rubber is proportional to temperature due to its entropy elasticity. In order to understand this behaviour thor- oughly, it is essential to understand the structure of rubber.

RESONANCE | February 2019 221 GENERAL ARTICLE

Box 2. When Does a Material Exhibit the Properties of Rubber?

• The material should be a high molecular weight polymer with an average molecular 5 5 weight (M¯ w)of5× 10 to 10 × 10 g/mol.

• On deformation, free rotation about the bonds provide high conformational possibilities.

• Long main chain facilitates inherent ﬂexibility.

• No aromatic rings, no conjugate double bonds, no bulky side groups, no polar groups, no hydrogen bonding.

• Small cohesive energy density.

• Amorphous.

• Lightly crosslinked.

• Glass transition temperature (Tg) is below room temperature.

4. Structure of Rubber

Rubber is composed of Rubber is composed of high molecular weight, chainlike, poly- high molecular weight, mer molecules. These are many thousand times larger than the chainlike, polymer low molecular weight conventional molecules. Weak London molecules. These are many thousand times forces act as the intermolecular forces between uncrosslinked rub- larger than the low ber molecules. But crosslinked rubber contains strong chemical molecular weight bonds. The atoms in the rubber chains (molecules) are in constant conventional molecules. thermal vibration. The average molecular weight (M¯ w) between crosslinks in vulcanised rubber is in the range of 5000–10000 g/mol. A material should satisfy certain molecular requirements in order to exhibit rubbery nature. These are summarized in Box 2. For more details on this topic, go through our earlier article [10]. In the main chain of rubber, the C–C–C bond angle is ﬁxed at 109.5o. But atoms have the freedom of rotation about the single bonds on the main chain and the side chains. Thus molecules

222 RESONANCE | February 2019 GENERAL ARTICLE possess many conformational possibilities in 3D space providing variable shapes to rubber molecules. The chain segments tend to coil up, instead of remaining linear. Hence, rubber molecules are extremely flexible, highly coiled chains with an entangled and convoluted structure. Due to this random structure, it possesses high entropy (disorder) in the normal state. This property along with the weak intermolecular forces allows rubber to deform greatly even under very small forces. However, as the rubber molecules are coiled and entangled, rubber possesses less flow and more stiffness.

5. Strain-Induced Crystallisation

Amorphous nature is one of the most important molecular requirements of rubber. Such a system has high entropy. But when rubber is stretched, its molecules are oriented in the direction of the applied force. Thus, the system becomes crystalline, and its entropy is reduced (Figure 2). This phenomenon is known as strain-induced crystallisation. The inter-chain and intra-chain forces which cause strain-induced crystallisation are not strong. Hence molecules regain the original state on releasing the applied force. This property depends upon the structure and regularity of Two observations the repeating units of rubber molecules and has significance in support strain-induced practical applications as it provides excellent mechanical proper- crystallisation. When a sample of unstretched ties and good resistance to crack growth in rubber products. Nat- rubber sample is frozen ural rubber, chloroprene rubber and polyisoprene rubber exhibit and is subsequently this property [11]. broken, the pieces take up irregular shape. But if How is this phenomenon confirmed? Two observations support a stretched rubber strain-induced crystallisation. When a sample of unstretched rub- sample is broken, ber sample is frozen and is subsequently broken, the pieces take parallel strands of fibres can be seen in the broken up irregular shape. But if a stretched rubber sample is broken, pieces. This dual nature parallel strands of fibres can be seen in the broken pieces. This of rubber can be dual nature of rubber can be observed with the naked eye. observed with the naked eye. The second observation pertains to the translucence of unstretched rubber and the cloudy appearance of stretched rubber. Unfilled rubber is amorphous. It can be considered as a homogeneous

RESONANCE | February 2019 223 GENERAL ARTICLE

medium without any obstacles which cause absorption or scat- Figure 2. Strain-induced tering. The transmission of light through the material is reason- crystallisation decreases the ably good, and hence it appears translucent. When it is stretched, entropy of natural rubber. molecules are oriented in the direction of the applied force. Thus crystallites are formed and, therefore, it becomes heterogeneous. Hence light is scattered, and rubber appears cloudy.

6. Thermodynamics in Engineering and Chemistry

The basic concept of thermodynamics is the transduction of heat into work and work into heat. It means that heat is the energy in transit corresponding to a deﬁnite amount of work. The word ‘thermodynamics’ has its origin in Greek: thermes meaning heat, and dynamikos meaning powerful or force [12].

In Nature, relative On rubbing our palms together, we can feel the warmth. Here me- motion (of surfaces) is chanical work/friction is converted into heat. But the heat gener- converted to heat due to ated thereby cannot be converted back to work. In Nature, relative friction. But the conversion of heat into motion (of surfaces) is converted to heat due to friction. But the work is not a natural conversion of heat into work is not a natural phenomenon. How- phenomenon. ever, a ‘heat engine’ transforms heat into work. Thermodynamics was developed as a subject in the late 18th and early 19th cen- turies to support man’s necessity to extract work from heat using heat engines [13]. But now thermodynamics is not limited to heat-work interconver-

224 RESONANCE | February 2019 GENERAL ARTICLE sion alone. It is used to explain diﬀerent phenomena/processes in diverse ﬁelds, such as information science [14, 15], economics [16] and biology [17] apart from mechanical engineering, chemical engineering [18], materials science, physics, chemistry [19], food science, etc. Also, it is interesting to note that areas like ‘human thermodynamics’ has emerged and evolved [20].

Though the basic concepts of thermodynamics are the same in Though the basic both engineering and chemistry, there are differences. One dif- concepts of ference is the sign conventions for work and heat. Work is de- thermodynamics are the same in both engineering fined as the action through a distance against an opposing force. and chemistry, there are It is the product of force and the distance to which the object is differences. One moved/displaced. Engineers give priority to the work a system difference is the sign can do. For example, steam in a heat engine can move a piston. conventions for work and heat. So in engineering (and physics), work is considered as positive (w>0) if it is done by the system. As a result, its internal energy falls. Also, work is negative, if it is done on the system. On the other hand, chemical systems (solids and liquids) do not undergo considerable change in volume during reactions. Hence no expansion work is observed in chemical systems, unlike gases. Chemists are more interested in energy changes associated with the system rather than that with the surroundings. Hence in chemistry, work done on the system is positive, and the work done by the system is negative. This results in an increase in internal energy as heat is transferred to the system as work. Heat gained by the system from the surroundings (endothermic process) is positive, whereas heat lost by the system to the surroundings (exother- mic process) is negative. It is interesting but confusing at times that different authors discuss the thermodynamics of rubber using both the conventions. This could be due to the interdisciplinary nature of polymer science and technology.

7. Elasticity of Metal and Rubber

Elasticity is the ability of a material to resist a deforming force and to restore its original size and shape when the force is re-

RESONANCE | February 2019 225 GENERAL ARTICLE

Figure 3. Rubber spring and metal spring.

moved. The higher the resistance to deformation, the greater is the elasticity of the material. A highly elastic material returns quickly to its original shape when the force is removed. Both metallic materials and rubber are elastic in nature. But the reasons for their elastic behaviuor are diﬀerent.

Since rubber undergoes Two parameters are used to measure the extent of elasticity – elas- much higher reversible tic limit and modulus. Elastic limit is the maximum stress that a deformations than material bears before it undergoes permanent deformation [21]. metallic materials, rubber can function at The elastic limit of metal is higher than that of rubber. Modu- high strains. Rubber can lus is the ratio of stress and strain. Metals undergo only a small also store elastic energy (≈1%) reversible strain, even under very large forces. Rubber, on about 150 times higher the other hand, is the most easily deformable material. It can be than that of steel and can release most of this stretched rapidly to very high elastic strains (500 to 1000%) even energy on retraction. under small loads. No other material is comparable to rubber as far as this property is concerned. Hence the modulus of elasticity of metal is higher than that of rubber. Also, on releasing the applied forces, rubber retracts rapidly and almost completely, with minimum energy loss. Since rubber undergoes much higher reversible deformations than metallic materials, rubber can function at high strains. Rubber can also store elastic energy about 150 times higher than that of steel and can release most of this energy on retraction. Therefore rubber dissipates less energy as heat during deformation. It means that rubber possesses small hysteresis loss (heat build-up). Hence a rubber spring can be a solid block whereas a metal spring acts by bending or twisting a long slender coiled structure (Figure 3). The deformation in metals can be of two types – plastic and elastic. Metals do not exhibit a change in volume during plastic defor-

226 RESONANCE | February 2019 GENERAL ARTICLE mation. But they show a deﬁnite change in volume during elastic deformation. The Poissons ratio is a measure of volume change during deformation. The Poissons ratio of materials like water, which does not exhibit a volume change is 0.5. Similarly, the Poisson’s ratio of rubber is also about 0.5. It means that the volume change of rubber is negligible during its deformation. But this ratio is about 0.35 for most of the metals. Another important diﬀerence is the energy elasticity of metal and the entropy elasticity of rubber. It is discussed in the next sec- tions.

8. Thermoelastic Experiment

Elasticity and Gough–Joule effect are described by the thermoe- Elasticity and lastic experiment. This experiment describes the unique thermal Gough–Joule effect are behaviour of rubber compared to metals. A metal strip elongates described by the thermoelastic on heating, while a piece of rubber shrinks. In the experiment, experiment. This thin pieces of metal and rubber are stretched under a load (W) experiment describes the within their elastic limits. The load should be small so as to avoid unique thermal permanent deformation. Choose a rubber that is capable of crys- behaviour of rubber compared to metals. tallising on stretching. In metals, the constituent atoms are held together in a 3D structure by the electrostatic attractive forces that exist between the positive ion core and the negative electron clouds. A metal strip elongates on loading (Figure 4), the application of force changes the size and shape of the atomic lattice. As a result, the internal energy of the system increases. This is an enthalpy (bond energy) effect. When the applied load is released, the lattice restores the original state by decreasing its internal energy. Hence the elasticity of metal is called energy-derived elasticity or energy elasticity.

When the stretched metal is heated, expansion occurs because of the increased oscillation of atoms about their equilibrium posi- tions. In other words, the eﬀects resulting from the stretching and heating of the metal are attributed to the intermolecular potential energy.

RESONANCE | February 2019 227 GENERAL ARTICLE

Figure 4. Thermoelastic experiment: Stretched metal elongates on heating.

9. Entropy Elasticity of Rubber

The conformation of Rubber elongates when a load W is applied to it (Figure 5). As rubber molecules a result, its molecules are uncoiled, and the entangled structure changes on deformation. is loosened. In the case of vulcanized rubber, chain segments But there is no change in bond length or bond uncoil till the crosslinks prevent uncoiling. The conformation angle. However, if the of rubber molecules changes on deformation. But there is no applied force is very change in bond length or bond angle. However, if the applied high, C–C bonds break force is very high, C–C bonds break causing the breakage of rub- causing the breakage of piece of rubber. ber piece. When a tensile load is applied, the rubber molecules align in the direction of force as mentioned earlier. The system becomes crystalline, and its entropy is reduced (Figure 2). This is a reversible process. When the applied force is released, free rotation of atoms causes molecules to coil up again. This happens due to the tendency of the system to maximize its entropy it had in its undeformed state. Hence the elastic memory of rubber is called entropy-derived elasticity or entropy elasticity. As neither bond stretching nor bond bending occurs during stretching, the change in enthalpy of rubber is zero. If we were able to stretch the rubber molecules, the rubber would have got straight- ened out. But a completely linear structure will not be obtained because the C–C–C bond angle is 109.5o and not 180o. Thus metallic material possesses energy elasticity, while an ideal rubber exhibits entropy elasticity. In order to be an ideal rubber,

228 RESONANCE | February 2019 GENERAL ARTICLE

Figure 5. Thermoelastic experiment: Stretched rubber shrinks on heating.

it should neither be uncrosslinked nor be highly crosslinked; it Metallic material should be optimally crosslinked. Also, the deformation of real possesses energy rubbers involves a change in volume. Hence in real rubbers, the elasticity, while an ideal rubber exhibits entropy elasticity is not based on entropy alone, it includes a change in elasticity. In order to be internal energy also. an ideal rubber, it should neither be uncrosslinked When stretched rubber is heated, molecules undergo chaotic move- nor be highly ment, and thus a random state is formed. The enhanced thermal crosslinked; it should be energy of the chains increases the molecular conformation and ac- optimally crosslinked. celerates its recovery. Hence under a constant load, rubber pieces shrink at high temperatures (Figure 5). In other words, the tension in stretched rubber increases on heating. The shrinkage due to the entropy of rubber is much higher than the thermal expansion of metals. For example, a rubber band contracts reasonably when heated under tension, but the elongation of a metal strip under a tensile load is not visible to the naked eye.

10. Origin of Elastic Force in Rubber

The internal energy of a system is due to its potential and kinetic energy. It is difficult to physically measure the internal energy of a system. But the change in internal energy (ΔU) of a system can be calculated from the initial and final values of the internal energy, i.e. Ufinal − Uinitial. According to the first law of thermodynamics, energy can neither be created nor be destroyed. Hence energy change of the system and the surroundings are related to each

RESONANCE | February 2019 229 GENERAL ARTICLE

The internal energy of a other as follows. system is due to its potential and kinetic energy. It is difficult to physically measure the ΔUsystem = −ΔUsurroundings internal energy of a ΔU + −ΔU = 0 . system. But the change system surroundings in internal energy of a system can be calculated A small change in the internal energy (dU) of an isolated system from the initial and final is due to two factors, i) heat exchange (dQ) between the system values of the internal W energy. and surroundings and ii) work done (d ) on the system or by the system. The first law of thermodynamics is represented by either of the following equations.

dU = dQ + dW (1)

dU = dQ − dW (2)

In (1), dU is the sum of total heat exchange into the system and the work done on the system by its the surroundings. Here the work done increases the internal energy of the system. When we use this equation, it means that work is done on the system. But in equation (2), dU of a closed system is equal to the heat supplied to the system, minus the work done by the system on its surroundings. These two equations are equivalent because,

Won the system = −Wby the system

Diﬀerent authors use either (1) or (2) when the thermodynamics of rubber is discussed [22, 23]. In this article, we follow (1). The second law of thermodynamics states that for a reversible process, dQ is equal to the product of entropy change (dS ) and absolute temperature (T).

dQ = TdS . (3)

230 RESONANCE | February 2019 GENERAL ARTICLE

Consider the stretching of a rubber band under a tensile force ( f ) Consider the stretching so as to cause a change in its length, dL. The work involved in of a rubber band under a stretching is a combination of work done by the system and the tensile force so as to cause a change in its work done on the system. The work done by the system (pressure- length. The work volume explansion) is negative. It is due to the expansion of rub- involved in stretching is ber against the atmosphere and is written as −PdV,whereP is the a combination of work external pressure and dV is the change in volume associated with done by the system and the work done on the elongation. But work done on the system (force-displacement system. work) is positive, and is written as + f dL.

dW = −PdV + f dL . (4)

Ideal rubber is an incompressible material without a change in volume. Hence the change in volume (of rubber network) on deformation is negligible, except for swelling. Therefore PdV is considered as zero here. But in more detailed analysis of elasticity, this term cannot be neglected. Hence,

dW = f dL . (5)

Substituting equations (5) and (3) in (1), we get

dU = TdS + f dL, (6)

f dL = dU − TdS. (7)

By taking the partial derivatives of the above equation with respect to length at constant temperature and volume, we get, ∂U ∂S f = − T . (8) ∂L T,V ∂L T,V

According to the above equation, a plot of force versus tempera- ∂S ture yields a straight line whose slope and intercept are ∂L T and ∂U ∂L T respectively. The elongation occurs as a result of tensile ∂U force involving only a change in the entropy. Hence ∂L T,V is negligible. Therefore, the elastic force for an ideal rubber is,

RESONANCE | February 2019 231 GENERAL ARTICLE

Figure 6. Thermoelas- tic experiment: Compressed gas expands on heating.

∂S f = −T . (9) ∂L T,V

The negative sign indicates that the force acts in a direction that is opposite to the increase in length. Also, it is clear that the force is proportional to the absolute temperature under a constant strain below 350%. This is true for an approximate range of temperatures from 200 to 400K.

11. Behaviour of an Ideal Gas

Consider the behaviour of an ideal (perfect) gas. Its pressure is due to the overall effect of impacts of atoms/molecules on the walls of the container. Gas molecules have kinetic energy. The internal energy is the sum of the kinetic energies of the individual gas molecules. A downward moving piston does work on a gas by compress- ing it. Gas molecules collide with the downward moving piston, and the pressure of the gas increases. This is caused neither by the intermolecular potential energy nor by the individual molecular kinetic energy. But it is due to the fewer configurations of molecules possible in the reduced volume. The effect of chang- ing the volume of a gas at constant temperature is, therefore, ex- plained on the basis of entropy and not energy. In a compressed

232 RESONANCE | February 2019 GENERAL ARTICLE gas, the extent of deformation is given by the reciprocal volume, 1 V (Figure 6). Increase in deformation (decrease in volume) corre- sponds to decrease in entropy. Therefore the pressure of an ideal gas is entropically derived. Also, on increasing the temperature of a compressed gas, the mo- mentum of its molecules increases. Hence when the compressed gas is heated, it expands to attain a state of maximum entropy. Consequently, energy is exchanged with the surroundings in the form of work. This property is the basis of heat engines. The equation of an ideal gas is obtained from (2).

dU = TdS − PdV. (10)

On rearranging the above equation,

PdV = TdS − dU. (11)

Diﬀerentiating the above equation with respect to volume, ∂S ∂U P = T − . (12) ∂V T ∂V T Here, we see two components to pressure similar to that of rubber – one is due to entropy, and the other is due to internal energy. According to Joules law, internal energy change is zero for an ideal gas. So the above equation becomes, ∂S P = T . (13) ∂V T ∂S Unlike rubber, ∂V T is positive for an ideal gas. Hence pressure acts in the same direction as increasing volume.

12. Analogies Between Rubber and Gas

Ideal gas and ideal rubber appear and behave diﬀerently. For example, ideal rubber is an incompressible material, unlike gases.

RESONANCE | February 2019 233 GENERAL ARTICLE

Ideal gas and ideal However, their thermodynamic and molecular behaviour are strik- rubber appear and ingly similar. behave diﬀerently. For example, ideal rubber is During an adiabatic process, no heat is absorbed or liberated. an incompressible This happens when the system is thermally insulated from the material, unlike gases. surroundings. In a mechanical system, this is achieved by a gas However, their thermodynamic and taken in a thermally insulated cylinder conﬁned by a piston. Rub- molecular behaviour are ber is an insulator. Hence no exchange of heat occurs across the strikingly similar. boundary during stretching and retraction of rubber. The stretching of an ideal rubber and the compression of an ideal gas are similar. Both cases involve a work input (work is done on the system) and a heat output. Adiabatic heating occurs when a gas is compressed (as in a diesel engine) due to work done on it by its surroundings. Similarly, adiabatic stretching of rubber increases its temperature. But the case of rubber is more com- plicated. If crystallisation occurs during stretching, part of the temperature increase is due to the latent heat of crystallisation. Adiabatic cooling occurs when the pressure of a gas is decreased rapidly (expansion), causing it to do work on its surroundings. Similarly, rapid retraction of rubber causes a decrease in temperature. The retractive force in rubber and decrease in pressure of gas are entropically derived. When rubber is stretched, its chainlike molecules uncoil, and its entropy decreases. Entropy-driven ‘pulling back’ can be sensed directly by touch. The reverse happens on contraction. Hence the retraction of rubber is sponta- neous similar to the free expansion of compressed gas. For both gas and rubber, the total entropy change for the reversible process is positive or zero since these act as (temporarily) isolated systems. Consider (8) for the elastic force in rubber and (12) for the pressure of a gas. In both cases there are two terms – one for the internal energy and the other for entropy. At constant temperature, the pressure is independent of internal energy of ideal gas (13). In analogy with this, the elastic force is independent of internal energy of an ideal rubber (9).

234 RESONANCE | February 2019 GENERAL ARTICLE

For an ideal gas, dW = −Popposing dV. During expansion, gas does work on the surroundings. Hence, work is negative by con- vention [24]. But since rubber is an incompressible material, the work for rubber becomes + f dL. This equation is analogous to the mechanical energy which is the product of force and distance. When rubber is stretched, work is done on the system and so it is positive.

13. Rubber as a Teaching Aid for Thermodynamics

The laws of thermodynamics can be applied to all systems such as solids, liquids and gases. If we change the temperature or pressure of solids and liquids, they do not undergo a noticeable change in their size; whereas the volume of a gas changes remarkably. Also, the use of real (non-ideal) gas leads to diﬃculties in solv- ing the equations of state. Hence in traditional physical chemistry textbooks, an ideal gas is used as the material to discuss the laws of thermodynamics [25, 26]. In mechanical/chemical engineering books, generally, steam is used as the material.

But measuring/handling the gases need complex instruments and Entropy is a difficult arrangements. Hence introductory lessons on thermodynamics thermodynamic concept are usually done without any classroom demonstrations; begin- to teach. However, ffi scientists have used ners find thermodynamics di cult. Teaching the concepts of ther- rubber to develop modynamics, therefore, necessitates the use of other materials, effective demonstration which behave like gases. As discussed above, the thermodynamic methods to introduce and molecular behaviour of ideal gas and ideal rubber are similar. entropy. Rubber can be used as the working Also, rubber can be handled easily in classroom demonstrations. substance in a heat Hence rubber is an alternative to gas in teaching thermodynamics engine, and rubber can [27–30]. also be used for lecture demonstration of Carnot Entropy is a difficult thermodynamic concept to teach. However, Cycle. scientists have used rubber to develop effective demonstration methods to introduce entropy [31–33]. The concept of heat engine has been discussed by Sarkar and Mondal [34], whereas uti- lization of rubber as the working substance in a heat engine has been described by other scientists [35–38]. While Srinivasan [39] has described Carnot’s research, scientists [40, 41] have also dis-

RESONANCE | February 2019 235 GENERAL ARTICLE

cussed the lecture demonstration of the Carnot Cycle using rubber. The teaching and learning of thermodynamics can be made easier by developing systematic teaching methodologies utilizing rubber as a teaching aid. Reader, now it is your turn; create a teaching aid using rubber appropriate for your subject.

14. Acknowledgement

Geetha is thankful to Prof. E Terentjev in the University of Cam- bridge, UK for providing a chance to study the actuation properties of rubber in his lab.