Question

Starting with \(\beta=(1 / V)(\partial V / \partial T)_{P},\) show that \(\rho=-(1 / \rho)(\partial \rho / \partial T)_{P},\) where \(\rho\) is the density.

Short answer

We are given the formula for the coefficient of isobaric volume expansion \(\beta\) and need to show the given equation involving density \(\rho\). We first write the relation between volume (V) and density as \(V = \frac{m}{\rho}\), and then differentiate both sides with respect to temperature (T) while keeping pressure (P) constant. This gives us: \(\frac{\partial V}{\partial T} = -\frac{m}{\rho^2} \frac{\partial \rho}{\partial T}\) Substituting the derivative of volume into the given formula for \(\beta\), we have: \(\beta = \frac{1}{V} \left(-\frac{m}{\rho^2} \frac{\partial \rho}{\partial T}\right)\) Then we substitute the relation between volume, mass, and density found in step 1 and simplify: \(\beta = -\rho \frac{1}{\rho^2} \frac{\partial \rho}{\partial T}\) Finally, canceling out one of the rho terms gives us the desired result: \(\rho = -\frac{1}{\rho} \frac{\partial \rho}{\partial T}\)

Step-by-step solution

Write the relation between volume and density

We know that: $$ \rho = \frac{m}{V} $$ Where \(\rho\) is the density, m is the mass, and V is the volume. We can rewrite this equation in terms of volume: $$ V = \frac{m}{\rho} $$

Differentiate the Volume and Density relation with respect to temperature

Now, we have to differentiate both sides of the equation with respect to temperature (T) while keeping the pressure (P) constant: $$ \frac{\partial V}{\partial T} = -\frac{m}{\rho^2} \frac{\partial \rho}{\partial T} $$

Relate the given formula with the differentiated equation

We are given the formula for the coefficient of isobaric volume expansion: $$ \beta= \frac{1}{V}\frac{\partial V}{\partial T} $$ We can substitute the expression for the derivative of the volume found in step 2: $$ \beta = \frac{1}{V} \left(-\frac{m}{\rho^2} \frac{\partial \rho}{\partial T}\right) $$

Substitute mass over density with volume and simplify

Now, we substitute the relation between volume, mass, and density found in step 1: $$ \beta = \frac{1}{\frac{m}{\rho}} \left(-\frac{m}{\rho^2} \frac{\partial \rho}{\partial T}\right) $$ And simplify the equation: $$ \beta = -\rho \frac{1}{\rho^2} \frac{\partial \rho}{\partial T} $$ Then, we can cancel out one of the rho terms to obtain the final result: $$ \rho = -\frac{1}{\rho} \frac{\partial \rho}{\partial T} $$

Key concepts

Density in Thermodynamics

Density, represented by \( \rho \), is a measure of mass per unit volume. It is a fundamental property in various sciences, including thermodynamics. In mathematical terms, it is expressed as:

• \( \rho = \frac{m}{V} \)

where \( m \) is mass and \( V \) is volume.
Understanding density is crucial as it helps explain how subjects like pressure and temperature influence a system's physical properties. In thermodynamics, changes in state variables such as temperature or pressure typically result in changes in both density and volume. Comprehending these changes is vital for solving complex problems in physics and engineering.
By expressing volume in terms of density, \( V = \frac{m}{\rho} \), it is possible to visualize how a change in temperature influences density—highlighting the inverse relationship between volume and density.

Partial Derivatives and Their Significance

Partial derivatives are a mathematical tool used to understand how a function changes with respect to one of several variables, keeping others constant. This concept is particularly useful in thermodynamics, where state properties depend on multiple variables.
Consider the partial derivative \( \left(\frac{\partial V}{\partial T} \right)_{P} \). It signifies the rate of change of volume with temperature at constant pressure. Such derivatives help describe the sensitivity and interdependence of physical quantities.
In thermodynamics, derivatives like \( \left(\frac{\partial \rho}{\partial T} \right)_{P} \) are crucial. For instance, this derivative assesses how density changes with temperature, providing insight into the behavior of materials under different thermal conditions.
By interpreting these derivatives, scientists and engineers can predict how substances react to environmental changes, aiding in the design of systems that efficiently manage thermal energy.

Volume-Temperature Relationship in Physics

In the context of thermodynamics, the volume-temperature relationship is pivotal. When the pressure is constant, changes in temperature directly impact the volume of a substance—a phenomenon described by the coefficient of isobaric volume expansion \( \beta \). This coefficient is defined as:

• \( \beta = \frac{1}{V}\left(\frac{\partial V}{\partial T} \right)_{P} \)

It quantifies how the volume of a material expands or contracts with temperature changes, assuming pressure remains the same.
Understanding this relationship is essential for assessing the behavior of gases, liquids, and solids under varying thermal conditions. Engineers consider \( \beta \) when designing materials and systems to ensure stability and integrity under temperature fluctuations.
Relating volume changes to temperature through \( \beta \) allows predictions of material behavior, ensuring practical applications can withstand environmental stresses.

Thermodynamics: The Science of Energy

Thermodynamics is the branch of physics focused on energy, work, and the laws governing their transformations. Central to this field are concepts like heat, temperature, and environmental interactions. Key principles of thermodynamics include:

• First Law: Energy cannot be created or destroyed, only transformed.
• Second Law: Systems naturally progress towards greater entropy.
• Third Law: As temperature approaches absolute zero, a system's entropy approaches a minimum value.

Understanding these laws allows scientists and engineers to harness energy effectively, designing processes that optimize the conversion between heat and work.
Thermodynamics provides a framework to analyze how energy flows in systems, from engines to natural ecosystems. By studying how variables like pressure, volume, and temperature interact, more efficient and sustainable solutions are developed.
This science is fundamental in multiple industries, driving innovations in energy efficiency, material science, and climate modeling.