Question

Derive an expression for the volume expansivity of a substance whose equation of state is $$P=\frac{R T}{v-b}-\frac{a}{v^{2} T}$$ where \(a\) and \(b\) are empirical constants.

Short answer

Question: Calculate the volume expansivity, β, from the following equation of state: \(P=\frac{R T}{v-b}-\frac{a}{v^{2} T}\). Answer: The volume expansivity, β, is given by the expression: $$\beta = \frac{\left(\frac{R}{v-b} - \frac{-a}{v^{2} T^{2}}\right)}{\frac{-RT}{(v-b)^{2}} - \frac{-2a}{v^3 T}}$$

Step-by-step solution

Definition of Volume Expansivity

The volume expansivity, also known as the coefficient of thermal expansion, is defined as the fractional change in volume per unit change in temperature at constant pressure: $$\beta = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P$$ Our goal is to find an expression for \(\beta\) in terms of the given equation of state.

Differentiate the Equation of State

We need to differentiate the given equation of state with respect to \(V\) and \(T\) at constant pressure. The given equation is: $$P=\frac{R T}{v-b}-\frac{a}{v^{2} T}$$ First, let's differentiate this equation with respect to \(V\) at constant pressure: $$0 = \left(\frac{\partial P}{\partial V}\right)_T = \frac{-RT}{(v-b)^{2}} - \frac{-2a}{v^3 T}$$ Now, let's differentiate the equation with respect to \(T\) at constant pressure: $$\frac{\partial P}{\partial T} = \frac{R}{v-b} - \frac{-a}{v^{2} T^{2}}$$

Solve for \(\frac{\partial V}{\partial T}\)

We will now use the two differentials that we just calculated to solve for the required \(\frac{\partial V}{\partial T}\). From the earlier definition of volume expansivity and the first differential, we have: $$\beta = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P = \frac{-1}{V}\frac{\frac{\partial V}{\partial T}}{\frac{\partial P}{\partial V}}$$ Using the second differential, we get: $$\frac{\partial V}{\partial T} = \frac{-V\frac{\partial P}{\partial T}}{\frac{\partial P}{\partial V}}$$ Substitute the values of \(\frac{\partial P}{\partial T}\) and \(\frac{\partial P}{\partial V}\) that we found in step 2: $$\frac{\partial V}{\partial T} = \frac{-V\left(\frac{R}{v-b} - \frac{-a}{v^{2} T^{2}}\right)}{\frac{-RT}{(v-b)^{2}} - \frac{-2a}{v^3 T}}$$

Find the Expression for Volume Expansivity

Now we will substitute the expression for \(\frac{\partial V}{\partial T}\) that we found in step 3 into the definition of volume expansivity to find the required expression for \(\beta\): $$\beta = \frac{1}{V}\left(\frac{-V\left(\frac{R}{v-b} - \frac{-a}{v^{2} T^{2}}\right)}{\frac{-RT}{(v-b)^{2}} - \frac{-2a}{v^3 T}}\right)$$ Simplify the expression: $$\beta = \frac{\left(\frac{R}{v-b} - \frac{-a}{v^{2} T^{2}}\right)}{\frac{-RT}{(v-b)^{2}} - \frac{-2a}{v^3 T}}$$ This is the expression for the volume expansivity \(\beta\) in terms of the given equation of state.

Key concepts

Equation of State

Understanding the equation of state is crucial for many fields of science, particularly in thermodynamics, where it provides a relationship between a substance's pressure (P), volume (V), and temperature (T). It's often presented as a mathematical equation characterizing the state of matter under a given set of physical conditions. For instance, the ideal gas law, given by the equation \(PV=nRT\), is a simple type of equation of state where P stands for pressure, V for volume, T for temperature, R is the universal gas constant, and n represents the amount of substance.

The equation of state provided in this problem incorporates empirical constants \(a\) and \(b\), which account for the interaction between particles and the volume excluded by them, respectively. The equation \(P=\frac{R T}{v-b}-\frac{a}{v^{2} T}\) must be carefully analyzed and differentiated appropriately to understand the substance's behavior upon changes in temperature, specifically looking at its volume expansivity.

Coefficient of Thermal Expansion

The coefficient of thermal expansion, often denoted by \(\beta\), is a material-specific value that indicates how much a substance's volume changes in response to a change in temperature. It's a fundamental concept in thermodynamics that allows engineers and scientists to predict the behavior of materials as they are heated or cooled.

In a practical sense, the coefficient of thermal expansion helps to understand and predict the structural integrity of materials under temperature variations, such as metals expanding on hot days and contracting on cold ones. Mathematically, we identify volume expansivity with the formula \(\beta = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P\), which says that \(\beta\) is directly proportional to the partial derivative of volume with respect to temperature, holding pressure constant. Highlighting its importance, this exercise aims to derive the coefficient of thermal expansion from an advanced equation of state, introducing a practical application of differential calculus in the process.

Differential Calculus in Thermodynamics

Differential calculus plays a pivotal role in thermodynamics, particularly when expressing how different properties of a system change relative to one another. It allows us to quantify the rate at which variables like volume and pressure change in response to variations in temperature and vice versa.

In this exercise, differential calculus is employed to find the partial derivatives involved in calculating the volume expansivity \(\beta\). The process requires an understanding of how to take partial derivatives of the equation of state with respect to volume and temperature while keeping pressure constant. Such differentiation provides the necessary information to solve for \(\beta\), illustrating the depth and utility of calculus within the field of thermodynamics. By mastering these calculations, students can apply these principles to more complex thermodynamic problems involving changes in state variables.