The equilibrium states of superheated steam are represented by Callendar's equation, thus: $$ \nu-b=\frac{r T}{P}-\frac{a}{T^{m}} \text { ? } $$ where \(b, r, a\), and \(m\) are constants. Calculate the volume expansivity \(\beta\) as a function of \(T\) and \(P\).

Callendar's equation is a thermodynamic relationship used to describe the behavior of superheated steam. It is represented as \(u - b = \frac{r T}{P} - \frac{a}{T^{m}}\). Here, \( b, r, a \), and \( m \) are constants specific to the steam and its properties.

This equation helps relate the specific volume \( u \), temperature \( T \), and pressure \( P \) of the steam. The constants \( b, r \) conduct adjustments for the gas properties, while \( a \) and \( m \) account for deviations from ideal behavior, especially at high temperatures (superheated conditions).

Callendar's equation is used in various applications, including power plant engineering and thermodynamic calculations, to predict how steam will behave under different conditions.

Superheated steam refers to steam that has been heated beyond its boiling point without increasing pressure. This process removes any liquid moisture, resulting in a higher temperature compared to boiling point temperature at the same pressure.

Superheated steam has essential applications, particularly in turbines and engines, because it can do more work and increases thermal efficiency.

Key points to consider about superheated steam:

• It increases the efficiency of thermodynamic cycles.
• It has higher energy content compared to saturated steam.
• Superheated steam prevents condensation in turbines, avoiding damage and loss of efficiency.

To study superheated steam, equations like Callendar's provide a more precise description, especially considering the non-ideal behavior of gases.

Differentiation is a fundamental concept in calculus that deals with the rate at which things change. For example, in our context, we differentiated the specific volume \( u \) with respect to temperature \( T \) to find volume expansivity \( \beta \).

Here's the step-by-step breakdown in our solution:

• Start from the rearranged Callendar's equation: \( u = \frac{r T}{P} - \frac{a}{T^{m}} + b \).
• Differentiate \( u \) with respect to temperature \( T \): \( \frac{du}{dT} = \frac{r}{P} + \frac{a m}{T^{m+1}} \).

Differentiation helps us understand how specific volume changes with temperature, aiding in the calculation of thermal properties like volume expansivity. Volume expansivity \( \beta \) expresses how a substance's volume changes with temperature, crucial for designing systems that involve heating or cooling.