Show that β=α(∂P / ∂T)v.

Name: Problem 48, Chapter 12: Thermodynamics: An Engineering Approach
Uploaded: 2020-09-26T10:02:00-08:00
Duration: 2 min 39 s
Channel: Sheh Lit Chang
Description: Show that β=α(∂P / ∂T)v.

Key Concepts

Thermal Expansion Coefficient

This coefficient measures the relative change in volume of a system per unit change in temperature at constant pressure. It provides an indication of how a material or system expands when heated, and is fundamentally defined by the partial derivative of volume with respect to temperature.

Thermal Pressure Coefficient

This coefficient quantifies how the pressure of a system changes with temperature when the volume is held fixed. It represents a system’s sensitivity to temperature variations in terms of pressure response and is often derived through interrelations with other thermodynamic coefficients.

Partial Derivative Relations in Thermodynamics

These relations involve understanding and applying the chain rule and cyclical relationships among variables. They allow us to connect different thermodynamic derivatives—for example, relating the change of pressure with temperature at constant volume to more basic quantities like the thermal expansion coefficient—thus establishing relationships between seemingly disparate properties.

Equation of State

An equation of state provides a functional relationship between thermodynamic variables such as pressure, volume, and temperature. It serves as a foundation for deriving various thermodynamic coefficients by expressing one variable in terms of others, enabling the extraction of partial derivatives that appear in expressions like the one equating the thermal pressure coefficient to the product of the thermal expansion coefficient and a derivative of pressure with respect to temperature.

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