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Chapter 1: Problem 30

Determine the expressions for the following, assuming that the ideal gas law holds. (a) \(\left(\frac{\partial n}{\partial V}\right)_{T, p}\) (b) \(\left(\frac{\partial T}{\partial p}\right)_{V, n}\) (c) \(\left(\frac{\partial n}{\partial T}\right)_{p, V}\) (d) \(\left(\frac{\partial p}{\partial n}\right)_{T, V}\)

Short Answer

Expert verified
(a) \( \frac{p}{RT} \); (b) \( \frac{V}{nR} \); (c) \( -\frac{pV}{RT^2} \); (d) \( \frac{RT}{V} \).

Step by step solution

01

Understanding the Ideal Gas Law

First, recall the ideal gas law equation, which is given by: \( pV = nRT \), where \( p \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature. We will use this equation to derive the expressions for each partial derivative requested in the problem.
02

Part (a): Derivative \( \left(\frac{\partial n}{\partial V}\right)_{T, p} \)

To find this derivative, rearrange the ideal gas law to solve for \( n \): \( n = \frac{pV}{RT} \). Taking the partial derivative with respect to \( V \) while keeping \( T \) and \( p \) constant, we have: \( \left(\frac{\partial n}{\partial V}\right)_{T, p} = \frac{p}{RT} \).
03

Part (b): Derivative \( \left(\frac{\partial T}{\partial p}\right)_{V, n} \)

Rearrange the ideal gas law to solve for \( T \): \( T = \frac{pV}{nR} \). The partial derivative with respect to \( p \) keeping \( V \) and \( n \) constant is: \( \left(\frac{\partial T}{\partial p}\right)_{V, n} = \frac{V}{nR} \).
04

Part (c): Derivative \( \left(\frac{\partial n}{\partial T}\right)_{p, V} \)

Using the rearranged form \( n = \frac{pV}{RT} \), take the partial derivative with respect to \( T \) while \( p \) and \( V \) are constant: \( \left(\frac{\partial n}{\partial T}\right)_{p, V} = -\frac{pV}{RT^2} \).
05

Part (d): Derivative \( \left(\frac{\partial p}{\partial n}\right)_{T, V} \)

Rearrange the ideal gas law for \( p \): \( p = \frac{nRT}{V} \). The partial derivative with respect to \( n \) keeping \( T \) and \( V \) constant is: \( \left(\frac{\partial p}{\partial n}\right)_{T, V} = \frac{RT}{V} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ideal Gas Law
The Ideal Gas Law is a fundamental equation in physics and chemistry that describes the relationship between pressure (\(p\)), volume (\(V\)), temperature (\(T\)), and the number of moles of gas (\(n\)). It is written as:

\[ pV = nRT \]where \(R\) is the ideal gas constant.
  • Pressure (\(p\)) refers to the force exerted by gas particles colliding with the walls of a container.
  • Volume (\(V\)) represents the space occupied by the gas.
  • Temperature (\(T\)) is the measure of the average kinetic energy of gas particles.
  • The number of moles (\(n\)) tells us how many molecules or atoms are present.
This equation is crucial for predicting how gases will behave under different conditions. It's an approximation that works best under low pressures and high temperatures.
Thermodynamics
Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. It involves the study of temperature, work, and the laws that govern energy transfer.
  • The First Law of Thermodynamics states that energy cannot be created or destroyed; it can only change forms, like from heat to work.
  • The Second Law tells us that the entropy, or disorder, of the universe tends to increase over time.
  • The Third Law explains that as the temperature of a system approaches absolute zero, its entropy approaches a minimum value.
The ideal gas law fits into the study of thermodynamics because it describes how gases exchange energy through pressure, volume, and temperature, which are fundamental thermodynamic variables.
Mathematical Derivation
When dealing with complex systems, mathematical derivations help us understand how variables are related. Here, partial derivatives are used to study how a single variable changes while others are held constant.
  • A partial derivative like \(\left(\frac{\partial n}{\partial V}\right)_{T, p}\) represents the rate of change of \(n\) with respect to \(V\), while keeping \(T\) and \(p\) constant.
  • By manipulating the ideal gas law, we find that \(n = \frac{pV}{RT}\) for a gas, allowing us to compute various partial derivatives.
  • These derivations are essential because they let us predict the behavior of a gas under varying conditions and are commonly used in physics and engineering.
Using derivatives in this way is part of calculus, which helps us understand rates of change within physical systems.
Physics Education
Physics Education aims to explain and explore complex physical concepts in an engaging and comprehensible manner. Using real-world examples, like the behavior of gases, enhances learning.
  • Interactive experiments, such as using balloons to demonstrate the ideal gas law, provide practical understanding.
  • Visual aids like videos and simulations can make abstract concepts more tangible, aiding retention and curiosity.
  • Conceptual teaching, focusing on understanding principles, is often prioritized over rote memorization.
Courses often use the ideal gas law within thermodynamics to bridge theoretical learning with experiential activities, deepening students' grasp of how scientific laws apply to the physical world around them. Understanding these concepts is crucial for students pursuing careers in science and engineering fields.

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Most popular questions from this chapter

A pot of cold water is heated on a stove, and when the water boils, a fresh egg is placed in the water to cook. Describe the events that are occurring in terms of the zeroth law of thermodynamics.

What does it mean to speak of a closed system? Give an example.

In the anaerobic oxidation of glucose by yeast, \(\mathrm{CO}_{2}\) is produced: $$ \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \rightarrow 2 \mathrm{CO}_{2}+2 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} $$ If \(1.56 \mathrm{~L}\) of \(\mathrm{CO}_{2}\) were produced at \(22.0^{\circ} \mathrm{C}\) and \(0.965 \mathrm{~atm}\), what mass of \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) is consumed by the yeast? Assume the ideal gas law applies.

For the following function, evaluate the derivatives in a-f below. $$ F(w, x, y, z)=3 x y^{2}+\frac{w^{3} z^{3}}{32 y}-\frac{x y^{2} z^{3}}{w} $$ (a) \(\left(\frac{\partial F}{\partial x}\right)_{w, y, z}\) (b) \(\left(\frac{\partial F}{\partial w}\right)_{x, y, z}\) (c) \(\left(\frac{\partial F}{\partial y}\right)_{w, x, z}\) (d) \(\left[\frac{\partial}{\partial z}\left(\frac{\partial F}{\partial x}\right)_{w, y, z}\right]_{w, x, y}\) (e) \(\left[\frac{\partial}{\partial x}\left(\frac{\partial F}{\partial z}\right)_{w, x, y}\right]_{w, y, z}\) (f) \(\left\\{\frac{\partial}{\partial w}\left[\frac{\partial}{\partial z}\left(\frac{\partial F}{\partial x}\right)_{w, y, z}\right]_{w, x, y}\right\\}_{x, y z}\)

What difference is necessary for heat to flow between two systems? Can you think of an exception to your answer?
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