Question

Several attempts have been made to express the partial derivatives of the most common thermodynamic properties in a compact and systematic manner in terms of measurable properties. The work of P. W. Bridgman is perhaps the most fruitful of all, and it resulted in the well-known Bridgman's table. The 28 entries in that table are sufficient to express the partial derivatives of the eight common properties \(P, T, v, s, u, h, f,\) and \(g\) in terms of the six properties \(P, v, T\) \(c_{p}, \beta,\) and \(\alpha,\) which can be measured directly or indirectly with relative ease. Obtain a copy of Bridgman's table and explain, with examples, how it is used.

Short answer

Answer: To use Bridgman's table to find the partial derivative of a thermodynamic property, first identify the desired derivative (e.g. \(\frac{\partial v}{\partial T}\)_P). Then, find an expression containing that derivative in Bridgman's table. In this case, the table provides \(\frac{\partial v}{\partial T}\)_P = vβ. Next, obtain the values for the measurable properties involved in the expression (specific volume and coefficient of volume expansion). Plug the given values into the expression to find the desired partial derivative. For example, with a specific volume of 0.8 m³/kg and a coefficient of volume expansion of 5 x 10⁻⁵/K, the partial derivative \(\frac{\partial v}{\partial T}\)_P = 4 x 10⁻⁵ m³/kg K.

Step-by-step solution

Understanding Bridgman's table

Bridgman's table lists a set of relations between partial derivatives of different thermodynamic properties, making it a useful resource in dealing with thermodynamic problems involving measurable properties. The eight commonly referenced properties are Pressure (P), Temperature (T), specific volume (v), entropy (s), internal energy (u), enthalpy (h), Helmholtz free energy (f), and Gibbs free energy (g). These can be expressed in terms of six measurable properties: P, v, T, heat capacity at a constant pressure (\(c_{p}\)), the coefficient of volume expansion (\(\beta\)), and the isothermal compressibility (\(\alpha\)).

Example problem - Simple explanation of how to use Bridgman's table

Let's suppose we are given a problem where we need to find the partial derivative \(\left(\frac{\partial v}{\partial T}\right)_P\) , i.e. the rate of change of the specific volume with temperature, given that the pressure is constant. We first look up Bridgman's table to find a relation containing the derivative we seek. Indeed, Bridgman's table includes the following expression: \(\left(\frac{\partial v}{\partial T}\right)_P = v\beta\) Where \(v\) is the specific volume and \(\beta\) is the coefficient of volume expansion. This expression is now in terms of measurable properties, which can be readily obtained either from the given problem or by calculation through other known properties.

Example problem - Using given values to find the partial derivative

Let's consider a specific example. Assume that the specific volume (v) of a substance is 0.8 m\(^3\)/kg and its coefficient of volume expansion (\(\beta\)) is 5 x 10\(^{-5}\)/K. We want to find \(\left(\frac{\partial v}{\partial T}\right)_P\). Using the expression from Bridgman's table: \(\left(\frac{\partial v}{\partial T}\right)_P = v\beta\) \(\left(\frac{\partial v}{\partial T}\right)_P = 0.8\,\text{m}^3\,\text{kg}^{-1} \times 5 \times 10^{-5}\,\text{K}^{-1}\) \(\left(\frac{\partial v}{\partial T}\right)_P = 4 \times 10^{-5}\,\text{m}^3\,\text{kg}^{-1}\,\text{K}^{-1}\) And with that, we have found the rate of change of specific volume with temperature while keeping the pressure constant using Bridgman's table.

Key concepts

Partial Derivatives in Thermodynamics

Partial derivatives play a fundamental role in thermodynamics as they enable us to understand how one thermodynamic property varies with another when all other variables are held constant. This is crucial for grasping how energy systems respond to changes in conditions, which is at the heart of designing and optimizing engines, refrigerators, and a myriad of real-world applications.

When delving into Bridgman's table, for instance, we deal with expressions like \(\left(\frac{\partial v}{\partial T}\right)_P\), which represents the rate of change of specific volume \(v\) as temperature \(T\) varies, while pressure \(P\) remains unchanged. This derivative, and others like it, can describe all manner of interactions within a system under specific constraints.

To make the most out of Bridgman's table, you must first understand what each symbol represents and the conditions of constant properties (e.g., subscript \(P\) meaning 'at constant pressure'). Then, locate the necessary partial derivative in the table to link it with the measurable properties provided or requested in your problem. The ability to interpret and utilize these derivatives is a key skill in thermodynamics.

Thermodynamic Properties

Thermodynamic properties can be broadly classified as either intensive or extensive. Intensive properties, such as temperature and pressure, do not depend on the quantity of matter in the system. Extensive properties, like volume and entropy, are dependent on the mass of the system. Understanding this distinction is critical when interpreting Bridgman's table, which allows various properties to be expressed in terms of each other.

Properties such as internal energy \(u\), enthalpy \(h\), and specific volume \(v\) give us insight into the energy content and state of a system. Bridgman's table elegantly combines these properties into relationships that can be quantified by derivatives, hence connecting various energy terms with measurable properties like heat capacity at constant pressure \(c_p\), coefficient of volume expansion \(\beta\), and isothermal compressibility \(\alpha\). These relationships are crucial for translating theoretical thermodynamics into practical measurements and applications.

Measurable Properties in Thermodynamics

Measurable properties in thermodynamics are the quantities that can be determined experimentally or observed directly. In the context of Bridgman's table, properties such as \(P\) (pressure), \(v\) (specific volume), \(T\) (temperature), \(c_{p}\) (heat capacity at constant pressure), \(\beta\) (coefficient of volume expansion), and \(\alpha\) (isothermal compressibility) are the foundational measurable properties that drive the table's usefulness.

Each of these properties is assessable through experimental means, such as pressure gauges, thermometers, calorimeters, or dilatometers. By accurately measuring these properties, you're equipped to use Bridgman's table for computing other thermodynamic quantities. For instance, knowing \(v\) and \(\beta\) allows you to determine how the specific volume of a substance will change with temperature, while pressure is kept constant, thus fulfilling a variety of intensive and extensive property calculations in an applied setting.