Consider the thermodynamic description of stretching rubber. The observables are the tension, \(\mathrm{t}\), and length, \(/\) (the analogues of \(p\) and \(\mathrm{V}\) for gases). Because \(d w=t d l\), the basic equation is \(d U=\mathrm{TdS}+\mathrm{tdl}\). (The term \(\mathrm{pdV}\) is supposed negligible throughout.) If \(G=U-T S-t l\), find expressions for \(\mathrm{d} G\) and dA, and deduce the Maxwell relations $$ \left(\frac{\partial S}{\partial l}\right)_{T}=-\left(\frac{\partial t}{\partial T}\right)_{l} \quad\left(\frac{\partial S}{\partial t}\right)_{T}=-\left(\frac{\partial l}{\partial T}\right)_{t} $$ Go on to deduce the equation of state for rubber,$$ \left(\frac{\partial U}{\partial l}\right)_{T}=t-\left(\frac{\partial t}{\partial T}\right)_{l} $$

The Gibbs free energy, often denoted by the symbol \(G\), is a fundamental thermodynamic quantity that reveals the amount of reversible work obtainable from a thermodynamic system at a constant temperature and pressure. For materials like rubber, understanding the Gibbs free energy is crucial for predicting their behavior under various conditions.

In the context of stretching rubber, the Gibbs free energy is defined as \(G = U - TS - tl\), where \(U\) is the internal energy, \(T\) is the temperature, \(S\) is the entropy, \(t\) is the tension, and \(l\) is the length. To find how the free energy changes (\(dG\)), we differentiate it with respect to its natural variables. In the given problem, because the term \(pdV\) is negligible, the differential form becomes \(dG = -SdT - ldt\). This expression is vital for thermodynamic considerations, such as determining the equilibrium state and understanding the reversibility of processes.

Practical Significance in Rubber

When analyzing the thermodynamics of rubber, the Gibbs free energy helps in determining the conditions under which the rubber can be stretched or compressed most efficiently. This has practical implications in manufacturing and materials engineering, especially when designing products that rely on the elastic properties of rubber, such as tires, gaskets, and elastic bands.

Maxwell relations are a set of equations derived from the second derivatives of thermodynamic potentials like the Gibbs free energy, which reflect the interdependence of thermodynamic variables. These relations are named after the renowned physicist James Clerk Maxwell.

The derivation of Maxwell relations is based on the symmetrical nature of partial derivatives. In thermodynamics, if a certain property has a continuous second derivative, the mixed derivatives taken in different orders are equivalent. From the differential forms of Gibbs free energy \(dG\) and Helmholtz free energy \(dA\), we deduce two Maxwell relations for the stretching of rubber:\begin{align*}\left(\frac{\partial S}{\partial l}\right)_T &= -\left(\frac{\partial t}{\partial T}\right)_l, \ \left(\frac{\partial S}{\partial t}\right)_T &= -\left(\frac{\partial l}{\partial T}\right)_t. \end{align*}These relations provide a deep insight into the thermodynamic behavior of rubber. For example, they tell us how the entropy change due to stretching is related to the tension response with temperature.

Implications of Maxwell Relations

Understanding Maxwell relations is key to relating measurable quantities, such as temperature changes due to stretching, to other thermodynamic variables. This is incredibly useful when dealing with complex materials like rubber, where direct measurements of certain properties may be challenging.

An equation of state is an equation that describes how state variables of a material, such as pressure, volume, and temperature, interrelate. It is often presented in a form that allows for prediction of the state of a material based on the conditions it experiences.

For rubber, the equation of state can be derived using thermodynamic principles and Maxwell relations. The derivation leads to the equation \(\left(\frac{\partial U}{\partial l}\right)_T = t - \left(\frac{\partial t}{\partial T}\right)_l\), which relates the internal energy change due to stretching (\(\partial U/\partial l\)) to measurable quantities like tension (\(t\)) and the tension's sensitivity to temperature (\(\partial t/\partial T\)). This equation is integral to understanding how rubber behaves under temperature variations when it's being stretched or compressed.

Real-World Application

The equation of state is particularly crucial in industrial processes that involve the mechanical treatment of rubber. It allows engineers to predict how the material will respond to different manufacturing settings, ensuring product durability and performance stability across varying operating temperatures.

The Helmholtz free energy, denoted by \(A\), is another thermodynamic potential that is especially useful for understanding systems at constant volume and temperature. For rubber, which can be considered a nearly constant-volume system when stretched, the Helmholtz free energy is defined as \(A = U - TS\), with \(U\) being the internal energy and \(S\) the entropy.

By differentiating the Helmholtz free energy, we obtain the differential form \(dA = tdl - SdT\), which shows how the energy available for work changes as the rubber is stretched (\(tl\)) and as the temperature changes (\(SdT\)). The behavior of \(A\) under different conditions is crucial when working with rubber in applications where volume changes are small but longitudinal tensile forces are significant.

Practical Insights from Helmholtz Free Energy

In practice, understanding the changes in Helmholtz free energy during the stretching of rubber can help in optimizing energy consumption and thermal efficiency in industrial processes. It also plays a role in the design and testing of rubber materials, ensuring that they meet the required specifications for mechanical and thermal performance in their end-use scenarios.