5) In thermodynamics, the ideal gas law states that PV = NKT Here • N,k are constants • P is the pressure of the gas • V is the volume of the container • T is the temperature. Notice that you can play with this equation in several ways. For example: • You can think of P as a function of V and T NKT P V. So you can ӘР ӘР via the equation find the partial derivatives ᎧᎢ ' Ꮩ • You can think of T as a function of P and V T = via the equation ƏT the partial derivatives OP¹ av • You can think of V as a function of P and T V via the equation ᎧᏙ find the partial derivatives OP¹ ƏT a) What do you think ap ar av ar av ap = dy writes dt rule? PV Nk. So you can find ƏT NKT P. So you can Ꮩ . will be equal to, based on a "naive" manipulation of the symbols? Similar to how "naively" one dy dx dx dt in the case of the chain b) What do you actually find if you do the product of these partial derivatives? [hint: there is a video on our Canvas sites: Thermodynamics and Partial Derivatives, a Cautionary Tale, where I go over this.]
5) In thermodynamics, the ideal gas law states that PV = NKT Here • N,k are constants • P is the pressure of the gas • V is the volume of the container • T is the temperature. Notice that you can play with this equation in several ways. For example: • You can think of P as a function of V and T NKT P V. So you can ӘР ӘР via the equation find the partial derivatives ᎧᎢ ' Ꮩ • You can think of T as a function of P and V T = via the equation ƏT the partial derivatives OP¹ av • You can think of V as a function of P and T V via the equation ᎧᏙ find the partial derivatives OP¹ ƏT a) What do you think ap ar av ar av ap = dy writes dt rule? PV Nk. So you can find ƏT NKT P. So you can Ꮩ . will be equal to, based on a "naive" manipulation of the symbols? Similar to how "naively" one dy dx dx dt in the case of the chain b) What do you actually find if you do the product of these partial derivatives? [hint: there is a video on our Canvas sites: Thermodynamics and Partial Derivatives, a Cautionary Tale, where I go over this.]
Chemistry
10th Edition
ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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Please assist me with homework question 5. Thanks in advance
![### Understanding the Ideal Gas Law and Partial Derivatives
#### Concept Overview
In thermodynamics, the ideal gas law is a fundamental equation that relates the pressure (P), volume (V), and temperature (T) of an ideal gas. The law is expressed as:
\[ PV = NkT \]
Where:
- \( N, k \) are constants,
- \( P \) is the pressure of the gas,
- \( V \) is the volume of the container,
- \( T \) is the temperature.
#### Exploring the Equation
Notice that you can manipulate this equation in several ways to gain different perspectives and insights. Here are a few approaches:
1. **Pressure as a function of Volume and Temperature**
\[ P = \frac{NkT}{V} \]
You can determine the partial derivatives:
\[
\frac{\partial P}{\partial T}, \quad \frac{\partial P}{\partial V}
\]
2. **Temperature as a function of Pressure and Volume**
\[ T = \frac{PV}{Nk} \]
You can determine the partial derivatives:
\[
\frac{\partial T}{\partial P}, \quad \frac{\partial T}{\partial V}
\]
3. **Volume as a function of Pressure and Temperature**
\[ V = \frac{NkT}{P} \]
You can determine the partial derivatives:
\[
\frac{\partial V}{\partial P}, \quad \frac{\partial V}{\partial T}
\]
#### Exercises
Consider the following questions:
**a) What do you think the product of these partial derivatives will be equal to, based on a "naive" manipulation of the symbols? Similar to how "naively" one might write:**
\[ \frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} \]
<img src="https://latex.codecogs.com/svg.latex?\Large&space;\frac{\partial P}{\partial T} \frac{\partial T}{\partial V} \frac{\partial V}{\partial P}">
**b) What do you actually find if you calculate the product of these partial derivatives?**
Hint: Refer to the video on our Canvas site titled: "Thermodynamics and Partial Derivatives, a Cautionary Tale," where](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F41a1af84-f79e-495e-9bf6-e155c46aa5b8%2F1dec8090-4341-4fe5-9a6d-481c181024f5%2Fotifanm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding the Ideal Gas Law and Partial Derivatives
#### Concept Overview
In thermodynamics, the ideal gas law is a fundamental equation that relates the pressure (P), volume (V), and temperature (T) of an ideal gas. The law is expressed as:
\[ PV = NkT \]
Where:
- \( N, k \) are constants,
- \( P \) is the pressure of the gas,
- \( V \) is the volume of the container,
- \( T \) is the temperature.
#### Exploring the Equation
Notice that you can manipulate this equation in several ways to gain different perspectives and insights. Here are a few approaches:
1. **Pressure as a function of Volume and Temperature**
\[ P = \frac{NkT}{V} \]
You can determine the partial derivatives:
\[
\frac{\partial P}{\partial T}, \quad \frac{\partial P}{\partial V}
\]
2. **Temperature as a function of Pressure and Volume**
\[ T = \frac{PV}{Nk} \]
You can determine the partial derivatives:
\[
\frac{\partial T}{\partial P}, \quad \frac{\partial T}{\partial V}
\]
3. **Volume as a function of Pressure and Temperature**
\[ V = \frac{NkT}{P} \]
You can determine the partial derivatives:
\[
\frac{\partial V}{\partial P}, \quad \frac{\partial V}{\partial T}
\]
#### Exercises
Consider the following questions:
**a) What do you think the product of these partial derivatives will be equal to, based on a "naive" manipulation of the symbols? Similar to how "naively" one might write:**
\[ \frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} \]
<img src="https://latex.codecogs.com/svg.latex?\Large&space;\frac{\partial P}{\partial T} \frac{\partial T}{\partial V} \frac{\partial V}{\partial P}">
**b) What do you actually find if you calculate the product of these partial derivatives?**
Hint: Refer to the video on our Canvas site titled: "Thermodynamics and Partial Derivatives, a Cautionary Tale," where
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