Problem 4 In the expression \((\partial u ... [FREE SOLUTION] (2024)

Open in App

Open in App Log out

Chapter 11: Problem 4

In the expression $(\partial u / \partial T)_{v}$, what does the subscript $v$signify?

Short Answer

Expert verified

The subscript $v$ signifies that the variable $v$ is held constant during the partial derivative.

Step by step solution

Identify the Expression Components

Consider the expression $\left(\frac{\partial u}{\partial T}\right)_{v}$. This expression represents a partial derivative.

Understand the Partial Derivative

In the given expression, $\left(\frac{\partial u}{\partial T}\right)_{v}$, $u$ is the dependent variable and $T$ is the independent variable. The role of the subscript needs to be understood.

Interpret the Subscript

The subscript $v$ indicates that the variable $v$ is held constant while taking the partial derivative of $u$ with respect to $T$.

Key Concepts

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

partial derivatives

Partial derivatives are a key concept in calculus, especially when dealing with functions that depend on multiple variables. In simple terms, a partial derivative measures how a function changes as one of its variables changes, keeping all other variables constant. Think of a situation where you have a 3D surface and you want to see how it changes if you move only along the x-axis, ignoring any changes along the y-axis.
In mathematical notation, the partial derivative of a function $ u = f(x, y) $ with respect to x, while holding y constant, is written as $ \frac{\partial u}{\partial x} $. This symbol tells you that you should focus only on the changes in the x direction, treating y as fixed.
Partial derivatives are widely used in various fields such as physics, engineering, and economics. They allow us to understand how different factors influence a system when they vary independently of one another.

thermodynamic variables

In thermodynamics, a branch of physics dealing with heat and temperature and their relation to other forms of energy and work, several variables are often considered. These are usually divided into two main categories:

Extensive Variables: These depend on the size or extent of the system. Examples include volume (V), internal energy (U), and mass.
Intensive Variables: These do not depend on the size of the system. Examples include temperature (T), pressure (P), and density.

Particular combinations of these variables are used to describe the state of the system, and changes in these states are often analyzed using partial derivatives.
For instance, $ \left(\frac{\partial u}{\partial T}\right)_v $ is used to describe how the internal energy (u) changes with temperature (T) when the volume (v) is held constant. By understanding these relationships, we can make predictions about how a system will behave under different conditions.

constant volume

The concept of constant volume is important in thermodynamic analysis, particularly when studying how different properties of a system change. When we see a partial derivative expression with a subscript, like $ \left(\frac{\partial u}{\partial T}\right)_v $, the subscript $ v $ signifies that the volume is held constant during the differentiation.
This makes sense in practical scenarios where the volume does not change, such as a gas trapped in a sealed container. Holding volume constant simplifies the analysis because it eliminates one variable from changing and affecting the result.
In summary, understanding partial derivatives with constant volume constraints helps us isolate the effects of temperature or other variables on a system. It makes our study more focused and manageable, giving us clearer insights into the thermodynamic properties.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

$Problem 4 In the expression \((\partial u ... [FREE SOLUTION] (3)$

Most popular questions from this chapter

During a phase change from liquid to vapor at fixed pressure, the temperatureof a binary nonazeotropic solution such as an ammonia-water solution increasesrather than remains constant as for a pure substance. This attribute isexploited in both the Kalina power cycle and in the Lorenz refrigerationcycle. Write a report assessing the status of technologies based on thesecycles. Discuss the principal advantages of using binary nonazeotropicsolutions. What are some of the main design issues related to their use inpower and refrigeration systems?For aluminum at $0^{\circ} \mathrm{C}, \rho=2700 \mathrm{~kg} /\mathrm{m}^{3}, \beta=71.4 \times 10^{-8}$ $(\mathrm{K})^{-1}, \kappa=1.34\times 10^{-13} \mathrm{~m}^{2} / \mathrm{N}$, and $c_{p}=0.9211 \mathrm{~kJ}/ \mathrm{kg} \cdot \mathrm{K} . \mathrm{De}-$ termine the percent error in$c_{v}$ that would result if it were assumed that $c_{p}=c_{v}$.A binary solution at $25^{\circ} \mathrm{C}$ consists of $59 \mathrm{~kg}$ ofethyl alcohol $\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)$ and $41\mathrm{~kg}$ of water. The respective partial molal volumes are $0.0573$ and$0.0172 \mathrm{~m}^{3} / \mathrm{kmol}$. Determine the total volume, in$\mathrm{m}^{3}$. Compare with the volume calculated using the molar specificvolumes of the pure components, each a liquid at $25^{\circ} \mathrm{C}$, inthe place of the partial molal volumes.A closed, rigid, insulated vessel having a volume of $0.142 \mathrm{~m}^{3}$contains oxygen $\left(\mathrm{O}_{2}\right)$ initially at 100 bar, $7^{\circ}\mathrm{C}$. The oxygen is stirred by a paddle wheel until the pressurebecomes 150 bar. Determine the (a) final temperature, in ${ }^{\circ} \mathrm{C}$. (b) work, in $\mathrm{kJ}$. (c) amount of exergy destroyed in the process, in $\mathrm{kJ}$. Let $T_{0}=7^{\circ} \mathrm{C}$.To investigate liquid-vapor phase transition behavior, construct a $p-v$diagram for water showing isotherms in the range \(0.7

See all solutions

Recommended explanations on Physics Textbooks

Force

Read Explanation

Nuclear Physics

Read Explanation

Electricity and Magnetism

Read Explanation

Fluids

Read Explanation

Oscillations

Read Explanation

Dynamics

Read Explanation

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free

This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept

Privacy & Cookies Policy

Privacy Overview

This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.

Necessary

Always Enabled

Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.

Non-necessary

Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.

SAVE & ACCEPT