**THE 1 ^{st} LAW OF THERMODYNAMICS**

**INTEGRAL FORMS OF 1st LAW**

We introduce here the following postulate of conservation of energy:

**the net rate at which energy flows into a body must be equal to the rate at which energy is accumulated in the body.**

This postulate can be put into the following mathematical form:

For a rigid body (or for a body undergoing a rigid motion) the work done results in accumulation of kinetic energy only (i.e. W_{t} = 0).

FOR A RIGID BODY:

** → there is no exchange between ‘thermal’and ‘kinetic’ energy!**

To deal with the thermodynamics of *real* materials, one needs to consider the deformation (e.g. change in density) of bodies. There are alternative forms of (eq 1a):

In the cases in which the body forces can be expressed as gradient of a scalar quantity which is independent of time:

(if *b**=-**gradΦ*, with *Φ *independent of time).

**LOCAL FORMS OF 1 ^{st} LAW**

Eqs. (1) can be integrated over any finite interval of time to obtain different * macroscopic formulations*. We can also obtain from the

From the Gauss theorem:

where * ρ* is the

which is a **local form of 1 ^{st} law.**

Now a question arises:

if

is it also true that ?

The answer is **YES, IF THE ‘PRINCIPLE OF LOCAL ACTION’ HOLDS TRUE.**

Moreover, if we introduce the following definition:

we consequently derive that:

Here *q* is a * local rate of heating per unit mass* and is related to

**THE 2 ^{nd} LAW OF THERMODYNAMICS**

**INTEGRAL FORMS OF 2 ^{nd} LAW**

The 2

In the 2

- is an extensive property;
- is a function of state.

The integral form of 2^{nd} law is expressed as:

where, by definition, *J _{t}* is the

1^{st} and 2^{nd} laws are coupled if, as postulated below, *J _{t}* is related to

By definition, we have:

and we * postulate* that:

where j is the* entropy flux* and

Based on eq. (6) and (7), the form of eq. (5a) for the 2^{nd} law becomes:

In writing 1^{st} and 2^{nd} law we have introduced four ‘primitive concepts’: * heat and energy* in the 1st law and

In the following we will introduce some special forms of the inequality (5).

- T not dependent on the point but, possibly, a function of time:

**ISOTHERMAL FORM **

- Adiabatic form (i.e.
*q=0; Q=0*), it holds for isolated systems:

**ADIABATIC FORM **

**LOCAL FORMS OF 2nd LAW**

From eq. (5c) we obtain the **local form of the 2 ^{nd} law:**

As for the case of U, if the principle of local action holds true,

Equation (8a) can be recast in the following form:

From the previous expression an alternative expression for the local form of the 2^{nd} law is obtained:

Also for the local form of 2^{nd} law we can introduce expressions for special cases.

- Case of
(i.e. ):**locally isothermal body**

**LOCALLY ISOTHERMAL BODY **

- Case of
(i.e. and hence q = 0):**steady state in a stationary body**

**STATIONARY BODY AT STEADY STATE **

From eq. (10) one can deduce that i**n the case of stationary heat conduction in stationary bodies the thermal flux vector must have a positive component in the direction of decreasing T.**

Eq. (8b) can be put in another form by considering that :

This expression is the basis for the dissipation theory. If we consider a process (σ_{t}=P(t) or σ = P (t)) at every time the 2^{nd} law (in each of the proposed forms) can be either satisfied as an * INEQUALITY* (in such a case the process is

For

In fact, multiplying eq. (8c) by T, we obtain:

where with and

We define now the **Helmholtz Free Energy, A:**

If we consider the * ISOTHERMAL CASE* (i.e. ):

Moreover, if it is also we have

Consequently:

**for isothermal reversible process, with T constant with time**

**for isothermal irreversible process, with T constant with time**

represents the **rate of accumulation of elastic energy.**

On the basis of previous expressions, in general, **for an isothermal process at constant T, the work is the sum of the non-thermal dissipation rate and of the rate of accumulation of elastic energy.**

*1*. Body, state and constitutive behaviour

*3*. 1st and 2nd principles of thermodynamics. Integral and local forms. The concept of entropy

*4*. State and equilibrium. Part 1

*5*. State and equilibrium. Part 2

*6*. State and equilibrium. Part 3

*7*. State and equilibrium. Part 4

Progetto "Campus Virtuale" dell'Università degli Studi di Napoli Federico II, realizzato con il cofinanziamento dell'Unione europea. Asse V - Società dell'informazione - Obiettivo Operativo 5.1 e-Government ed e-Inclusion