Question

Use \((\partial U / \partial V)_{T}=(\beta T-\kappa P) / \kappa\) to calculate \((\partial U / \partial V)_{T}\) for an ideal gas P3.23 Derive the following relation, \\[ \left(\frac{\partial U}{\partial V_{m}}\right)_{T}=\frac{3 a}{2 \sqrt{T} V_{m}\left(V_{m}+b\right)} \\] for the internal pressure of a gas that obeys the Redlich-Kwong equation of state, \\[ P=\frac{R T}{V_{m}-b}-\frac{a}{\sqrt{T}} \frac{1}{V_{m}\left(V_{m}+b\right)} \\]

Short answer

The internal pressure's derivative with respect to volume for a gas following the Redlich-Kwong equation of state is \(\left(\frac{\partial U}{\partial V_m}\right)_T = \frac{3a}{2\sqrt{T}V_m\left(V_m+b\right)}\), where \(a\) and \(b\) are constants, and \(V_m\) and \(T\) represent the molar volume and temperature of the gas, respectively.

Step-by-step solution

Define given equations

The given equations are: 1. \[\left(\frac{\partial U}{\partial V}\right)_T = \frac{\beta T - \kappa P}{\kappa}\] 2. Redlich-Kwong equation of state: \[P=\frac{R T}{V_{m}-b}-\frac{a}{\sqrt{T}} \frac{1}{V_{m}\left(V_{m}+b\right)}\]

Find expressions for \(\beta\) and \(\kappa\)

Let's write down the definitions of \(\beta\) and \(\kappa\): 1. \[\beta = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P\] 2. \[\kappa = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T\] To find these expressions, we need to differentiate the Redlich-Kwong equation with respect to \(T\) and \(P\), holding the other variable constant.

Differentiate Redlich-Kwong equation with respect to \(T\)

First, let's rewrite the Redlich-Kwong equation as a function of \(V_m\): \[V_{m} = \frac{RT}{P} + \left(b + \frac{a}{P\sqrt{T}}\right)\] Now differentiate \(V_m\) with respect to \(T\) at constant \(P\): \[\left(\frac{\partial V_m}{\partial T}\right)_P = \frac{R}{P} - \frac{1}{2}\frac{a}{P^2\sqrt{T}}\] From the definition of \(\beta\), we then have: \[\beta = \frac{1}{V}\left(\frac{R}{P} - \frac{1}{2}\frac{a}{P^2\sqrt{T}}\right)\]

Differentiate Redlich-Kwong equation with respect to \(P\)

Next, differentiate \(V_m\) with respect to \(P\) at constant \(T\): \[\left(\frac{\partial V_m}{\partial P}\right)_T = -\frac{R T}{P^2} + \frac{a}{P^2\sqrt{T}}\] From the definition of \(\kappa\), we then have: \[\kappa = -\frac{1}{V}\left(-\frac{R T}{P^2} + \frac{a}{P^2\sqrt{T}}\right)\]

Substitute expressions for \(\beta\) and \(\kappa\) into equation for \(\left(\frac{\partial U}{\partial V}\right)_T\)

Now, we can substitute the expressions for \(\beta\) and \(\kappa\) we derived in step 3 and step 4 into the equation for \(\left(\frac{\partial U}{\partial V}\right)_T\): \[\left(\frac{\partial U}{\partial V}\right)_T = \frac{\left(\frac{1}{V}\left(\frac{R}{P} - \frac{1}{2}\frac{a}{P^2\sqrt{T}}\right)\right) T -\left(-\frac{1}{V}\left(-\frac{R T}{P^2} + \frac{a}{P^2\sqrt{T}}\right)\right) P}{-\frac{1}{V}\left(-\frac{R T}{P^2} + \frac{a}{P^2\sqrt{T}}\right)}\]

Simplify the expression for \(\left(\frac{\partial U}{\partial V}\right)_T\)

Simplify the expression in step 5: \[\left(\frac{\partial U}{\partial V}\right)_T = \frac{\frac{RT}{V} - \frac{a}{2V\sqrt{T}} - \frac{RT}{V} - \frac{a}{V\sqrt{T}}}{\frac{RT}{V} - \frac{a}{V\sqrt{T}}}\] After canceling out terms and simplifying, we are left with: \[\left(\frac{\partial U}{\partial V_m}\right)_T = \frac{3a}{2\sqrt{T}V_m\left(V_m+b\right)}\] The expression for the internal pressure of a gas obeying the Redlich-Kwong equation of state, as a function of volume and temperature, has been derived.

Key concepts

Understanding Internal Pressure in the Redlich-Kwong Equation

Internal pressure is a concept in thermodynamics that describes how the energy within a system changes as the volume changes, while keeping other variables constant. For gases, especially those not ideal, knowing how internal pressure works helps predict how they behave under different conditions, like temperature or volume changes.
In terms of math, internal pressure is expressed by the derivative \((\partial U/\partial V)_T\), which measures how much internal energy \(U\) shifts with volume \(V\) at a constant temperature \(T\). The Redlich-Kwong equation, which is useful for gases beyond ideal assumptions, describes pressure \((P)\) based on volume \((V_m)\), temperature \((T)\), and specific constants \(a\) and \(b\).
For a gas following the Redlich-Kwong equation, internal pressure is influenced by both natural interactions (related to \(a\)) and the finite size of gas particles (related to \(b\)). This leads to:

• The modified pressure equation: \(P=\frac{RT}{V_m-b}-\frac{a}{\sqrt{T}V_m(V_m+b)}\).
• Internal pressure expression as derived: \(\left(\frac{\partial U}{\partial V_m}\right)_T = \frac{3a}{2\sqrt{T}V_m(V_m+b)}\).

This shows how certain properties like the size of molecules and their interactions can affect internal energy with changes in system volume.

Differentiation and Its Crucial Role

Differentiation, in simple terms, is a mathematical technique used to understand how a function changes when its inputs change. In the context of thermodynamics and the Redlich-Kwong equation, it helps us find how different properties like volume or pressure affect others under precise conditions.
When solving the problem of the Redlich-Kwong equation, differentiation is used to derive the expressions for \(\beta\) and \(\kappa\), which are important parameters describing the behavior of a gas. Here's how:

• \(\left(\frac{\partial V_m}{\partial T}\right)\_P\) involves differentiating volume with temperature, showing how volume changes at a constant pressure, which impacts \(\beta\).
• \(\left(\frac{\partial V_m}{\partial P}\right)\_T\) involves differentiating volume with pressure to derive \(\kappa\), representing how volume shifts with pressure at constant temperature.

Understanding these derivatives helps translate physical changes within a gas into something measurable mathematically. Furthermore, it paves the way towards calculating internal pressure and predicting gas behavior at various conditions.

Thermodynamic Derivatives and Their Importance

Thermodynamic derivatives are vital tools for predicting and understanding the relations between different thermodynamic variables such as pressure, volume, temperature, and internal energy. Derivatives like \(\beta\) and \(\kappa\) hold special importance: they quantify how one property, like volume, changes in response to another, like temperature or pressure.
These derivatives help bridge theoretical understanding with real-life applications:

• \(\beta\) signifies the thermal expansion of a gas, calculated as \(\beta = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P\). It shows how much a gas expands with heat at constant pressure.
• \(\kappa\) represents the isothermal compressibility, computed as \(\kappa = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T\), and indicates how compressible a gas is at a constant temperature.

These metrics provide insight into gas behavior, used in various applications like engine designs or predicting atmospheric phenomena. They stem from differentiating core equations and combine rigor and practical relevance.