Question

Another equation of state is the Berthelot equation, \(V_{m}=(R T / P)+b-\left(a / R T^{2}\right) .\) Derive expressions for \(\beta=1 / V(\partial V / \partial T)_{P}\) and \(\kappa=-1 / V(\partial V / \partial P)_{T}\) from the Berthelot equation in terms of \(V, T,\) and \(P\).

Short answer

The expressions for the coefficient of thermal expansion (\(\beta\)) and the isothermal compressibility (\(\kappa\)) using the Berthelot equation of state are as follows: \(\beta = \frac{1}{V}\cdot M\left(\frac{R}{P} + \frac{2a}{RT^3}\right)\) \(\kappa = \frac{1}{V} \cdot M\frac{RT}{P^2}\)

Step-by-step solution

Express \(V\) in terms of \(V_m\)

We have the relation \(V = V_m n\), where n is the number of moles. We can rewrite this equation as follows: \(V = \frac{Vm}{M}\), where \(M\) is the molar mass. Now, we will use the Berthelot equation to substitute \(V_m = V\cdot M\) in our expressions for \(\beta\) and \(\kappa\).

Differentiate the Berthelot equation with respect to \(T\) at constant \(P\)

Our goal is to find \(\left(\frac{\partial V}{\partial T}\right)_P\) in order to find \(\beta\). We can take the derivative of the Berthelot equation with respect to \(T\) while keeping \(P\) constant: \(\frac{\partial V}{\partial T} = M\cdot\frac{\partial V_m}{\partial T}\) \(\frac{\partial V_m}{\partial T}\) = \(\frac{\partial}{\partial T}\left(\frac{RT}{P} + b - \frac{a}{RT^2}\right)\) From this, we can calculate \(\left(\frac{\partial V}{\partial T}\right)_P\): \(\left(\frac{\partial V}{\partial T}\right)_P = M\left(\frac{R}{P} + \frac{2a}{RT^3}\right)\)

Calculate \(\beta\)

Now that we have \(\left(\frac{\partial V}{\partial T}\right)_P\), we can find the expression for \(\beta\): \(\beta = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P\) \(\beta = \frac{1}{V}\cdot M\left(\frac{R}{P} + \frac{2a}{RT^3}\right)\)

Differentiate the Berthelot equation with respect to \(P\) at constant \(T\)

Our goal is to find \(\left(\frac{\partial V}{\partial P}\right)_T\) in order to find \(\kappa\). We can take the derivative of the Berthelot equation with respect to \(P\) while keeping \(T\) constant: \(\frac{\partial V}{\partial P} = M\cdot\frac{\partial V_m}{\partial P}\) \(\frac{\partial V_m}{\partial P}\) = \(\frac{\partial}{\partial P}\left(\frac{RT}{P} + b - \frac{a}{RT^2}\right)\) From this, we can calculate \(\left(\frac{\partial V}{\partial P}\right)_T\): \(\left(\frac{\partial V}{\partial P}\right)_T = -M\left(\frac{RT}{P^2}\right)\)

Calculate \(\kappa\)

Now that we have \(\left(\frac{\partial V}{\partial P}\right)_T\), we can find the expression for \(\kappa\): \(\kappa = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T\) \(\kappa = -\frac{1}{V}\cdot\left(-M\frac{RT}{P^2}\right)\) \(\kappa = \frac{1}{V} \cdot M\frac{RT}{P^2}\) Finally, we have expressions for \(\beta\) and \(\kappa\): \(\beta = \frac{1}{V}\cdot M\left(\frac{R}{P} + \frac{2a}{RT^3}\right)\) \(\kappa = \frac{1}{V} \cdot M\frac{RT}{P^2}\)

Key concepts

Thermodynamic Equations of State

Thermodynamic equations of state are mathematical models that describe how the physical properties of a system change with conditions. One common example is the Berthelot equation, which is an attempt to improve upon the ideal gas law by accounting for intermolecular forces. The Berthelot equation gives the molar volume of a gas, denoted by \(V_m\), as a function of pressure \(P\), temperature \(T\), and specific constants \(a\) and \(b\) that relate to the particular gas in question.

Understanding this equation helps in studying the behavior of real gases, especially under varying conditions of temperature and pressure. It forms the basis for calculating other valuable thermodynamic properties, such as isothermal compressibility and the coefficient of thermal expansion, by using partial differentiation techniques.

Partial Differentiation in Thermodynamics

In thermodynamics, partial differentiation is a mathematical tool used to understand how a particular property changes with one variable while keeping other variables constant. For instance, deriving expressions for isothermal compressibility and thermal expansion coefficients involves differentiating the equation of state with respect to pressure and temperature, respectively.

The Berthelot equation's derivatives with respect to \(T\) holding \(P\) constant, and \(P\) holding \(T\) constant, provide insights into how volume changes under these specific conditions. These partial derivatives are foundational in calculating two essential thermodynamic properties: \(\beta\) (coefficient of thermal expansion) and \(\tau\) (isothermal compressibility).

Isothermal Compressibility

Isothermal compressibility, denoted by \(\tau\), is a measure of a material's relative volume change under small pressure changes while maintaining a constant temperature. For a gas following the Berthelot equation, we calculate \(\tau\) using partial differentiation to analyze how pressure affects volume at a constant temperature.

The calculation from the Berthelot equation involves differentiating with respect to \(P\) at a constant \(T\) and then dividing by the negative of the volume \(V\). This process highlights the intertwined nature of different thermodynamic variables and the need for a solid understanding of differential calculus in thermodynamics. The significance of \(\tau\) lies in its ability to predict how a substance responds to pressure changes, which is crucial in designing pressure vessels and understanding the behavior of substances under various conditions.

Coefficient of Thermal Expansion

The coefficient of thermal expansion, represented by the symbol \(\beta\), quantifies how the size of a material changes with temperature. In the context of the Berthelot equation, \(\beta\) describes how the volume of a gas changes when the temperature changes at a constant pressure.

The calculation involves partially differentiating the Berthelot equation with respect to temperature \(T\) while keeping pressure \(P\) constant and then dividing by the volume \(V\). This coefficient is critical for understanding and predicting the effects of thermal processes on materials, including the design of structures or systems that will experience temperature variations, ensuring their reliability and integrity over a range of temperatures.