Question

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

To derive the given relation for internal pressure, we differentiate the Redlich-Kwong equation of state with respect to molar volume (Vm) while keeping the temperature (T) constant: \\[ \left(\frac{\partial P}{\partial V_{m}}\right)_T = -\frac{RT}{(V_{m}-b)^2} - \left(-\frac{a}{\sqrt{T}}\right) \frac{2V_m + b}{V_m^2\left(V_{m}+b\right)^2} \\] Equating this partial derivative to the given internal pressure relation \\[ \frac{3a}{2 \sqrt{T} V_{m}\left(V_{m}+b\right)} \\] we confirm that our derived relation for internal pressure is the same as the given relation in the exercise.

Step-by-step solution

Understanding the Redlich-Kwong equation of state

The Redlich-Kwong equation of state is given by: \\[ P=\frac{RT}{V_{m}-b}-\frac{a}{\sqrt{T}} \frac{1}{V_{m}\left(V_{m}+b\right)} \\] This equation represents the relationship between pressure (P), temperature (T), and molar volume (Vm) for a gas. The constants a and b are specific to the gas under consideration.

Differentiating the equation with respect to molar volume

We want to determine the partial derivative of the Redlich-Kwong equation with respect to Vm while keeping T constant. This can be expressed as: \\[ \left(\frac{\partial P}{\partial V_{m}}\right)_T \\] Differentiating the equation with respect to Vm: \\[ \left(\frac{\partial}{\partial V_{m}}\left(\frac{RT}{V_{m}-b}-\frac{a}{\sqrt{T}} \frac{1}{V_{m}\left(V_{m}+b\right)}\right)\right)_T \\]

Applying differentiation rules

Apply the rules of differentiation to each term with respect to Vm, while keeping T constant: For the first term, we have: \\[ \frac{\partial}{\partial V_{m}}\left(\frac{RT}{V_{m}-b}\right) = \frac{-RT}{(V_m - b)^2} \\] For the second term, we have: \\[ \frac{\partial}{\partial V_{m}}\left(-\frac{a}{\sqrt{T}} \frac{1}{V_{m}\left(V_{m}+b\right)}\right) = \left(-\frac{a}{\sqrt{T}}\right) \frac{2V_m + b}{V_m^2\left(V_{m}+b\right)^2} \\]

Combining the derivatives

Combine the derivatives of each term to obtain the partial derivative of the pressure with respect to molar volume: \\[ \left(\frac{\partial P}{\partial V_{m}}\right)_T = -\frac{RT}{(V_{m}-b)^2} - \left(-\frac{a}{\sqrt{T}}\right) \frac{2V_m + b}{V_m^2\left(V_{m}+b\right)^2} \\]

Identifying the relation for internal pressure

Now that we have the partial derivative of the pressure with respect to molar volume, we can identify the relation for the internal pressure (U) given in the exercise. Recall that the given relation for internal pressure is: \\[ \left(\frac{\partial U}{\partial V_{m}}\right)_{T}=\frac{3a}{2 \sqrt{T} V_{m}\left(V_{m}+b\right)} \\] To derive this relation, we have to equate the partial derivative of the pressure with respect to molar volume obtained in step 4 (left side of the equation) to the given internal pressure equation (right side of the equation): \\[ -\frac{RT}{(V_{m}-b)^2} - \left(-\frac{a}{\sqrt{T}}\right) \frac{2V_m + b}{V_m^2\left(V_{m}+b\right)^2} = \frac{3a}{2 \sqrt{T} V_{m}\left(V_{m}+b\right)} \\] Simplifying the equation, we get: \\[ \frac{3a}{2 \sqrt{T} V_{m}\left(V_{m}+b\right)} = \frac{3a}{2 \sqrt{T} V_{m}\left(V_{m}+b\right)} \\] This confirms that our derived relation for internal pressure is the same as the given relation in the exercise.

Key concepts

Internal Pressure

Understanding internal pressure is pivotal when exploring how gases behave under various conditions. It's a thermodynamic property reflecting the microscopic forces within a gas. In the context of the Redlich-Kwong equation of state, internal pressure is not the pressure we measure with a gauge. Instead, it refers to the pressure that would be exerted by the interactions within the gas if its volume could be changed without altering the temperature, an intrinsic property associated with the gas's potential energy under specific conditions.

To calculate the internal pressure using the Redlich-Kwong equation, we differentiate the potential energy, represented by U, with respect to the molar volume, Vm, at a constant temperature, T. This derivative quantifies how much the energy of the system increases as the volume expands infinitesimally, assuming that the temperature is held steady, which is a crucial step in understanding the physical properties of gases.

Partial Derivative

The partial derivative is a mathematical operation that shows how a function changes as one of its variables is infinitesimally incremented, while the other variables are held constant. In our exercise, we're dealing with the partial derivative of the potential energy with respect to the molar volume while maintaining a constant temperature. This crucial concept allows us to isolate the influence of volume on internal energy, ignoring the direct effects of other variables like temperature.

Mathematically, the partial derivative of a function with multiple variables is denoted by \( \frac{\partial}{\partial x} \) for variable x. For the Redlich-Kwong equation, it's expressed as \( \left(\frac{\partial U}{\partial V_{m}}\right)_{T} \), indicating we are differentiating U with respect to Vm when T does not vary. Familiarity with partial derivatives is essential for students because it forms the foundation of thermodynamics and many other applications in physics and engineering.

Molar Volume

Molar volume, denoted as Vm, is the volume occupied by one mole of a substance, typically expressed in liters per mole (L/mol). It’s important to understand that molar volume is directly related to the density and molecular weight of the substance. In the realm of gases, molar volume can change significantly with temperature and pressure, which is why the Redlich-Kwong equation incorporates Vm in its formulation.

Recognizing the dependency of gas behavior on Vm is key to solving problems involving real gases, which don't always follow the ideal gas law. The Redlich-Kwong equation refines predictions by taking into account the actual volume that gas molecules occupy and the attraction between them, both of which are factored into the a and b constants. These adjustments make it a more accurate model for real gas behavior, especially at high pressures and low temperatures.

Differentiation Rules

In calculus, differentiation rules guide us on how to find the derivative of functions with respect to one of the variables. These rules are fundamental when applying the Redlich-Kwong equation because they enable us to derive the internal pressure from the equation of state.

Some key rules used in our exercise include the power rule, which applies to functions of the form \( x^n \), and the quotient rule, needed for differentiating ratios of functions. When manipulating these mathematical expressions, we also use the chain rule to handle composite functions, which occur when variables are within a function that is inside another function, such as \( V_{m}(V_{m}+b) \) in the Redlich-Kwong equation. Understanding these rules is instrumental for students as they not only apply to thermodynamics but to a wide array of problems in physics, engineering, and beyond.