Book edition 1st

Author(s) Daniel V. Schroeder

Pages 356 pages

ISBN 9780201380279

Answers without the blur.

Just sign up for free and you're in.

Short Answer

The formula for $C_{P} - C_{V}$ derived in the previous problem can also be derived starting with the definitions of these quantities in terms of U and H. Do so. Most of the derivation is very similar, but at one point you need to use the relation $P = - (\partial F / \partial V)_{T}$ .

$(C_{P} - C_{V}) = (T {(\frac{\partial S}{\partial V})}_{T} {(\frac{\partial V}{\partial T})}_{P})$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Given

$(C_{P} - C_{V})$

Explanation

The specific heat of a substance can be of two types:(i) specific heat at constant pressure C_P(ii) specific heat at constant volume C_V

They are given by

$C_{V} = {(\frac{\partial U}{\partial T})}_{V} C_{P} = {(\frac{\partial H}{\partial T})}_{P}$

Where, U = internal energy, H = enthalpy, V = volume and P = pressure.

Lets consider U=U(V, T), and differentiate above expression with respect to T, we get
$d U = {(\frac{\partial U}{\partial V})}_{T} d V + {(\frac{\partial U}{\partial T})}_{V} d T$

Similarly write expression for enthalpy H=U+P V.

Write the expression for d H at constant pressure d H=d U+P d V

Substitute $d U = ({(\frac{\partial U}{\partial V})}_{T} d V + {(\frac{\partial U}{\partial T})}_{V} d T)$

We get,

$d H = {(\frac{\partial U}{\partial V})}_{T} d V + {(\frac{\partial U}{\partial T})}_{V} d T + p d V = [{(\frac{\partial U}{\partial V})}_{T} + P] d V + {(\frac{\partial U}{\partial T})}_{V} d T$

Simplify, divide both sides of the above expression by d T,

${(\frac{\partial H}{\partial T})}_{P} = [{(\frac{\partial U}{\partial V})}_{T} + P] {(\frac{\partial V}{\partial T})}_{P} + {(\frac{\partial U}{\partial T})}_{V} . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)$

Substitute $C_{P} f o r {(\frac{\partial H}{\partial T})}_{P} a n d C_{V} f o r {(\frac{\partial U}{\partial T})}_{V}$

$C_{P} = [{(\frac{\partial U}{\partial V})}_{T} + P] {(\frac{\partial V}{\partial T})}_{P} + C_{V} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)$

Substitute $- {(\frac{\partial F}{\partial V})}_{T} f o r P$ in equation (2)

$C_{P} = {(\frac{\partial (U - F)}{\partial V})}_{T} {(\frac{\partial V}{\partial T})}_{P} + C_{V}$

Now substitute TS for (U-F)

$C_{P} = T {(\frac{\partial S}{\partial V})}_{T} {(\frac{\partial V}{\partial T})}_{P} + C_{V^{\infty}}$

So, the value of $(C_{P} - C_{V}) = (T {(\frac{\partial S}{\partial V})}_{T} {(\frac{\partial V}{\partial T})}_{P})$

Find Study Materials for

Create Study Materials

Select your language

An Introduction to Thermal Physics

Answers without the blur.

Short Answer

The formula for $C_{P} - C_{V}$ derived in the previous problem can also be derived starting with the definitions of these quantities in terms of U and H. Do so. Most of the derivation is very similar, but at one point you need to use the relation $P = - (\partial F / \partial V)_{T}$ .

Step by Step Solution

Given

Explanation

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Select your language

An Introduction to Thermal Physics

Answers without the blur.

Short Answer

The formula for CP-CV derived in the previous problem can also be derived starting with the definitions of these quantities in terms of U and H. Do so. Most of the derivation is very similar, but at one point you need to use the relation P=-(∂F/∂V)T.

Step by Step Solution

Given

Explanation

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Work Energy and Power

Radiation

Astrophysics

Physics of Motion

Scientific Method Physics

Famous Physicists

Fluids

Torque and Rotational Motion

Nuclear Physics

Fields in Physics

Measurements

Space Physics

Electricity

Physical Quantities and Units

Medical Physics

Electrostatics

Further Mechanics and Thermal Physics

Mechanics and Materials

Engineering Physics

Energy Physics

Force

Particle Model of Matter

Waves Physics

Magnetism

Modern Physics

Atoms and Radioactivity

Dynamics

Electricity and Magnetism

Conservation of Energy and Momentum

Fundamentals of Physics

Kinematics Physics

Circular Motion and Gravitation

Linear Momentum

Rotational Dynamics

Oscillations

Translational Dynamics

Geometrical and Physical Optics

Thermodynamics

Magnetism and Electromagnetic Induction

Electromagnetism

Electric Charge Field and Potential

Turning Points in Physics

The formula for $C_{P} - C_{V}$ derived in the previous problem can also be derived starting with the definitions of these quantities in terms of U and H. Do so. Most of the derivation is very similar, but at one point you need to use the relation $P = - (\partial F / \partial V)_{T}$ .