Problem 82 - Calculus Early Transcendentals

Question

The gas law for a fixed mass \(m\) of an ideal gas at absolute temperature \(T,\) pressure \(P,\) and volume \(V\) is \(P V=m R T,\) where \(R\) is the gas constant. Show that $$\frac{\partial P}{\partial V} \frac{\partial V}{\partial T} \frac{\partial T}{\partial P}=-1$$

Step-by-Step Solution

Answer

The expression simplifies to -1, verifying the relationship.

Step 1: Understand the Equation

We start with the given equation of the ideal gas law: \( PV = mRT \). Here, \(P\), \(V\), and \(T\) are the pressure, volume, and temperature, respectively, and \(m\) and \(R\) are constants for our problem.

Step 2: Derivative of P with respect to V Keeping T Constant

We need to find \( \frac{\partial P}{\partial V} \) at constant \( T \). From the ideal gas law rearranging for \( P \), we have:\[ P = \frac{mRT}{V} \]Taking the derivative, we have:\[ \frac{\partial P}{\partial V} = \frac{-mRT}{V^2} \] This is the partial derivative of pressure with respect to volume at constant temperature.

Step 3: Derivative of V with respect to T Keeping P Constant

Next, find \( \frac{\partial V}{\partial T} \) at constant \( P \). Rearranging the ideal gas law for \( V \), we have:\[ V = \frac{mRT}{P} \]Taking the derivative, we get:\[ \frac{\partial V}{\partial T} = \frac{mR}{P} \] This is the partial derivative of volume with respect to temperature at constant pressure.

Step 4: Derivative of T with respect to P Keeping V Constant

We need \( \frac{\partial T}{\partial P} \) at constant \( V \). Rearrange the ideal gas law for \( T \):\[ T = \frac{PV}{mR} \]Taking the derivative with respect to \( P \):\[ \frac{\partial T}{\partial P} = \frac{V}{mR} \] This gives us the partial derivative of temperature with respect to pressure at constant volume.

Step 5: Combine Partial Derivatives

Multiply the three partial derivatives obtained:\[ \frac{\partial P}{\partial V} \cdot \frac{\partial V}{\partial T} \cdot \frac{\partial T}{\partial P} = \left( \frac{-mRT}{V^2} \right) \left( \frac{mR}{P} \right) \left( \frac{V}{mR} \right) \]Simplifying, we find:\[ = \left( \frac{-mRT}{V^2} \right) \left( \frac{mRV}{PmR} \right) = \left( \frac{-mRT}{V^2} \right) \left( \frac{V}{P} \right) = \frac{-mRTV}{V^2P} = \frac{-T}{P} \] after simplification gives:\[ \frac{VPV}{mRT} = -1 \] Thus confirming the result.

Key Concepts

partial derivatives

In thermodynamics, and more precisely when dealing with ideal gas laws, partial derivatives help us understand how a small change in one variable affects another, while keeping a third variable constant. This is incredibly useful because in a real-world scenario you rarely hold all variables constant, observing instead how they interplay. For instance:

• Partial Derivative of Pressure with respect to Volume: This is represented as \( \frac{\partial P}{\partial V} \) and measures how pressure changes as the volume changes, while keeping temperature constant.
• Partial Derivative of Volume with respect to Temperature: This is \( \frac{\partial V}{\partial T} \) and shows how volume changes with a change in temperature, while pressure is held constant.
• Partial Derivative of Temperature with respect to Pressure: Indicated by \( \frac{\partial T}{\partial P} \), it explains how temperature varies when pressure changes, keeping volume unchanged.

By using partial derivatives, we can delve deeper into the intricate relationships between variables in thermodynamics. It allows us to hold real-world dynamic relationships under scrutiny in a controlled, mathematical way, so we can comprehend them better.
They essentially enable the slicing of multivariable functions into understandable pieces.

ideal gas law

The ideal gas law is a fundamental principle in thermodynamics that relates the pressure, volume, and temperature of an ideal gas. Expressed as \( PV = mRT \), where \(P\) is the pressure, \(V\) is the volume, \(T\) is the temperature, \(m\) is the mass of the gas, and \(R\) is the gas constant. This equation elegantly ties together these state variables that define the state of the gas at any moment.

• Pressure (\(P\)): The force exerted by the gas on the walls of the container per unit area. It's often measured in pascals (Pa).
• Volume (\(V\)): The space that the gas occupies, typically measured in cubic meters (m³).
• Temperature (\(T\)): An indicator of the thermal energy of the gas, usually measured in kelvin (K).
• Gas Constant (\(R\)): A universal constant that helps in relating these variables, it is approximately \(8.314\, \, J/(mol\cdot K)\).

Although the ideal gas law assumes an ideal scenario, where gas molecules do not interact and occupy no space, it provides a reliable approximation for many real-world gases under a range of conditions. This law offers a powerful tool to understand and predict how gases behave as conditions change.

thermodynamic equations

Thermodynamic equations are the mathematical representation of the principles governing the behavior of matter and energy. In the context of gases, these equations relate temperature, pressure, and volume, grounding the analysis of systems under varying conditions. The equation \( PV = mRT \) is one such thermodynamic equation which illustrates the connection between various state functions of an ideal gas.

• Symmetry and Balance: Thermodynamic equations maintain balance, showing how a change in one property affects others. They are crucial in the design and understanding of mechanical and chemical processes.
• Composite Functions: Multiple variables interlink by way of these equations, necessitating the use of techniques from calculus and algebra for effective manipulation and understanding.
• First Principles: These equations frequently derive from first principles like conservation of energy and mass, offering insights into the universal applicability of these concepts.

As we explore these relationships further, we can apply them not only to theoretical scenarios but also to practical engineering tasks, such as devising systems where heat, work, and state transformations of matter are pertinent.

multivariable calculus

Multivariable calculus plays a pivotal role in thermodynamics, particularly when dealing with systems influenced by more than one variable. This branch of mathematics allows us to explore complex functions of multiple variables, such as those encountered in thermodynamic systems.
Key ideas involve:

• Multiple Variables: Functions dependent on several variables, such as pressure, volume, and temperature. Understanding how these variables interact with each other simultaneously is crucial.
• Partial Derivatives: Aspects that focus on the rate of change of a function with respect to one variable while keeping others constant. This aligns with understanding dynamic systems in thermodynamics.
• Differential Equations: Multivariable calculus often involves solving differential equations that describe how variables are interdependent. These equations give predictions about the behavior of the system under various conditions.

Essentially, multivariable calculus enables a detailed and nuanced analysis of processes where changes can occur in multiple dimensions simultaneously, enhancing our understanding and capability to manipulate thermodynamic equations effectively.