Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Transcription for Educational Use
#### Given Equation
\[
\left( P + \frac{n^2 a}{V^2} \right) (V - nb) = nRT
\]
#### Assumptions
Assuming \( n, a, b, R, T \) are constants, find \(\frac{dP}{dV}\) using implicit differentiation.
#### Steps for Implicit Differentiation
1. Differentiate both sides of the equation with respect to \( V \):
\[
\left( \frac{dP}{dV} \cdot (V - nb) + P - \frac{n^2 a}{V^2} \left( \frac{-2n^2 a}{V^3} \right) \right) + \left( \frac{dn}{dV} \right) (V + s {n} \frac{v^s {n}} 10)
\]
2. Simplified expression after differentiation:
\[
(0 - 1)x \left(\frac{-n^2}{n^(1/5)} + g \right) + \left( \frac{dn}{dV} \right) \left(n^{e/}{V^5 n^s n s - q} \right)
\]
(Note: Some symbols and constants like \( g, q, e, s \), etc., are assumed based on context and may need verification for accuracy in the specific domain).
#### Explanation
This section explains the implicit differentiation method on the given thermodynamic equation, which is a form of the Van der Waals equation for real gases. By assuming certain constants, this process involves finding the rate at which pressure changes concerning the change in volume, which is a common problem in thermodynamics and calculus.
#### Notes
No graphs or diagrams were present in the provided content. The transcription focuses solely on the handwritten mathematical expressions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a10fbfc-d7d7-4914-b7d7-e6510260e301%2F29fe764c-904e-4855-9ca5-4e529619f3f4%2Ft7grvg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Transcription for Educational Use
#### Given Equation
\[
\left( P + \frac{n^2 a}{V^2} \right) (V - nb) = nRT
\]
#### Assumptions
Assuming \( n, a, b, R, T \) are constants, find \(\frac{dP}{dV}\) using implicit differentiation.
#### Steps for Implicit Differentiation
1. Differentiate both sides of the equation with respect to \( V \):
\[
\left( \frac{dP}{dV} \cdot (V - nb) + P - \frac{n^2 a}{V^2} \left( \frac{-2n^2 a}{V^3} \right) \right) + \left( \frac{dn}{dV} \right) (V + s {n} \frac{v^s {n}} 10)
\]
2. Simplified expression after differentiation:
\[
(0 - 1)x \left(\frac{-n^2}{n^(1/5)} + g \right) + \left( \frac{dn}{dV} \right) \left(n^{e/}{V^5 n^s n s - q} \right)
\]
(Note: Some symbols and constants like \( g, q, e, s \), etc., are assumed based on context and may need verification for accuracy in the specific domain).
#### Explanation
This section explains the implicit differentiation method on the given thermodynamic equation, which is a form of the Van der Waals equation for real gases. By assuming certain constants, this process involves finding the rate at which pressure changes concerning the change in volume, which is a common problem in thermodynamics and calculus.
#### Notes
No graphs or diagrams were present in the provided content. The transcription focuses solely on the handwritten mathematical expressions.
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