Chapter 2, Problem 2.3PR

(a)

Interpretation Introduction

Interpretation:

The variation of internal energy of a non-metallic solid at very low temperature $\text{(T}\text{=}\text{0)}$ has to be calculated.

Concept introduction:

Internal energy:

Internal energy of a system is the total energy contained in the system.  It keeps an account for the loss and gain of energy of the system due to changes in internal state.  It is dependent on temperature and pressure.

It is denoted as $\text{U}$.

1^st law of thermodynamics:

The 1^st law of thermodynamics states that the total energy of the isolated system is constant and that can be transformed into one form to another like heat to work or vice versa but can neither be created nor be destroyed.

It is represented as,

$\text{ΔU}\text{=}\text{W}\text{+Q}$

Where,

$\text{ΔU}\text{=}$ Change in internal energy of the system

W is the energy transferred as the form of work to the system.

Q is the energy transferred as the form of heat to the system

Heat capacity at constant volume:

Specific heat capacity at constant volume is defined as the amount of heat required to increase the temperature by ${\text{1}}^{\text{ο}}\text{C}$ keeping the system at constant volume.

It is denoted as ${\text{C}}_{\text{v}}$

${\text{C}}_{\text{v}}\text{=}{\left(\frac{\partial \text{U}}{\partial \text{T}}\right)}_{\text{v}}$

$\begin{array}{l}\text{U}\text{=}\text{internal}\text{energy}\text{of}\text{the}\text{system}\\ \\ \text{T}\text{=}\text{temperature}\text{of}\text{the}\text{system}\end{array}$

(b)

Interpretation Introduction

Interpretation:

The expression of molar specific heat at constant volume has to be interpreted at temperature $\text{T}$.

Concept introduction:

Internal energy:

Internal energy of a system is the total energy contained in the system.  It keeps an account for the loss and gain of energy of the system due to changes in internal state.  It is dependent on temperature and pressure.

It is denoted as $\text{U}$.

Internal energy per mole is called molar internal energy that is denoted by ${\text{U}}_{\text{m}}$.

1^st law of thermodynamics:

The 1^st law of thermodynamics states that the total energy of the isolated system is constant and that can be transformed into one form to another like heat to work or vice versa but can neither be created nor be destroyed.

It is represented as,

$\text{ΔU}\text{=}\text{W}\text{+Q}$

Where,

$\text{ΔU}\text{=}$ Change in internal energy of the system

W is the energy transferred as the form of work to the system.

Q is the energy transferred as the form of heat to the system

Heat capacity at constant volume:

Specific heat capacity at constant volume is defined as the amount of heat required to increase the temperature by ${\text{1}}^{\text{ο}}\text{C}$ keeping the system at constant volume.

It is denoted as ${\text{C}}_{\text{v}}$

Specific heat capacity at constant volume per mole is considered as molar specific heat capacity at constant volume which is denoted as ${{\text{(C}}_{\text{v}}\text{)}}_{\text{m}}$

${\text{C}}_{\text{v}}\text{=}{\left(\frac{\partial \text{U}}{\partial \text{T}}\right)}_{\text{v}}$

$\begin{array}{l}\text{U}\text{=}\text{internal}\text{energy}\text{of}\text{the}\text{system}\\ \\ \text{T}\text{=}\text{temperature}\text{of}\text{the}\text{system}\end{array}$