The equation of state of an ideal gas is \(P V=n R T\), where \(n\) and \(R\) are constants. (a) Show that the volume expansivity \(\beta\) is equal to \(1 / T\). (b) Show that the isothermal compressibility \(\kappa\) is equal to \(1 / P\).

Volume expansivity, denoted as \(\beta\), describes how the volume of a substance changes as the temperature changes, under constant pressure. This is crucial in understanding how gases expand when heated. Mathematically, we define volume expansivity as: \[ \beta = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_{P} \] where \(V\) is the volume and \(T\) is the temperature. Using the Ideal Gas Law, which is given by \[ P V = n R T, \] we can differentiate with respect to temperature at constant pressure to derive \(\beta\). After substituting and solving, we find that \[ \beta = \frac{1}{T}. \] This indicates that the volume expansivity of an ideal gas is inversely proportional to the temperature.

Isothermal compressibility, denoted as \( \kappa \), measures how the volume of a gas changes as the pressure changes, under constant temperature. It's a concept used when compressing gases is considered, such as in pneumatic systems. Isothermal compressibility is defined as: \[ \kappa = -\frac{1}{V} \left( \frac{\partial V}{\partial P} \right)_{T}. \] When we take the Ideal Gas Law, \[ P V = n R T, \] and differentiate with respect to pressure at constant temperature, we can isolate \( \left( \frac{\partial V}{\partial P} \right)_{T} \) and substitute into our definition of \( \kappa \). This results in the finding that \[ \kappa = \frac{1}{P}. \] This shows that the isothermal compressibility of an ideal gas is inversely proportional to the pressure.

Understanding differentiation in thermodynamic equations helps in deriving relationships between variables like pressure, volume, and temperature. When dealing with thermodynamic problems, we often need to differentiate equations to find how one variable affects another while some factors stay constant.
For the Ideal Gas Law \[ P V = n R T, \] there are two main differentiation scenarios:
- At constant pressure: We differentiate with respect to temperature to find changes in volume for a given temperature increase.
- At constant temperature: We differentiate with respect to pressure to find volume changes due to pressure variations.
Applying these methods, we can derive vital thermodynamic properties like volume expansivity and isothermal compressibility, aiding in our understanding of gas behavior.

The Ideal Gas Constant, symbolized by \( R \), is a fundamental constant in the Ideal Gas Law \[ P V = n R T. \] It ensures the consistent relationship between pressure, volume, and temperature for an ideal gas. The Ideal Gas Constant has a value of approximately 8.314 J/(mol·K).
Key points include:

• It connects microscopic properties to macroscopic properties of gases.
• It allows for the calculation of a gas's internal energy, enthalpy, and other thermodynamic properties.
• Its universality ensures that the same gas law applies to all ideal gases, simplifying calculations and derivations.

Understanding \( R \) is essential for studying gas dynamics, heat engines, and thermodynamics as a whole.