Question

The theory of rubber elasticity and the theory of ideal gas dynamics show that the two equations, G = nRT and PV = n'RT, share certain common thermodynamic ideas.

What are they?

Answer 1

Significance of volume changes.

states that in an extension carried out at constant volume (by suitable adjustment of the surrounding pressure)
the internal energy change should be zero. The implication is that the changes of internal energy which are actually observed in an extension at constant pressure arise from the accompanying change of volume. Following up this conclusion, and assuming the classical relation between internal energy and volume for a pure dilatation to be applicable, Gee derived an expression for the total change of volume in an extension from an unstrained length lG to a length 1, at constant pressure,namely,

where K is the volume compressibility. Direct measurements of volume changes during extension (Gee et al 1950), though somewhat inaccurate, were found to be consistent with this equation.

Anisotropic compressibility. Both Gee and Elliott and Lippman assumed the
rubber sample, acted on by a tensile forcef, to be isotropically compressible under a
superimposed hydrostatic pressure, that is, that the changes in linear dimensions
are the same in all directions. Thus aV/V = 3W1, or

The more rigorous theory of Flory, on the other hand, leads to an anisotropic compressibility in the strained state, the form of which is represented by the relation

where a: is the extension ratio (defined later). This formula, which, as Flory notes, is substantially the same as that previously derived on an essentially similar basis by Khasanovich (1959), reduces to (4.12) as the extension (a-1) tends to zero, representing the fact that for sufficiently small strains the rubber may be considered to be isotropic.

Intramolecular internal energy.

Gee’s provisional conclusion, namely that the changes in internal energy on extension arise solely from the accompanying changes of volume, implies that any such internal energy changes are to be associated with the forces between the polymer molecules (Van der Waals forces), these forces being of the same kind as those which determine the volume of an ordinary lowmolecular- weight liquid. Flory, however, recognized the need for the inclusion of an internal energy term associated with the conformation of the polymer molecule itself, and arising from energy barriers to rotation about bonds within the single chain. This leads to the possibility of a finite internal energy contribution to the stress, even under constant volume conditions.
In Flory’s derivation the force-extension relation (2.10) given by the elementary gaussian network theory is replaced by the formula

in which v is the total number of chains in the network (the chain being defined as before as the molecular segment between successive cross-linkages), Zi is the length of the undistorted specimen corresponding to the volume V in the strained state(li = V1’3), and 01 = l/li, where 1 is the strained length. The additional factor ~ f i r :is the ratio of the mean square length $ of the network chains in the undistorted state of the network (at volume V ) to the mean square length 3 of an identical set of free chains.The significance of the factor e/z will be apparent on recalling that in the elementary theory it was assumed that = 2, that is, that the chains in the unstrained state of the network had the same mean square length as a corresponding set of free chains (cf $2). Even if this assumption should happen to be valid at one particular temperature (and this cannot be demonstrated), it would still not apply at any other, since is determined by the volume of the network, which depends on temperature, while 2 is, of course, an independent statistical property of the free or unrestricted chain, which will in general also be temperature dependent. Differentiation of equation (4.14) under conditions corresponding to constant volume and to constant pressure, respectively, leads to the following expressions for the corresponding changes in internal energy (or heat content)

from which we obtain

This equation enables the stress-temperature coefficient at constant volume to be derived from experimental data on the stress-temperature coefficient under the normal constant pressure conditions.It is convenient to represent the quantity (2UjaZ),,, the internal energy contribution to the force at constant volume, by the symbol fe. The fractional contribution of the internal energy to the force, at constant volume, on the basis of equation is therefore

Flory shows further that experiments at constant pressure and constant extension ratio are not equivalent to experiments at constant volume and constant length, as proposed by Gee ; the correct relation between the stress-temperature coefficients under these respective conditions is.

Hence the following Properties discussed above are the common thermodynamic ideas .