Chapter 4, Problem 4.1E

List the sets of conditions that allow $dS$, $dU$, and $dH$ of a process in a system act as a spontaneity condition.

Interpretation Introduction

Interpretation:

The sets of conditions that allow $dS$, $dU$ and $dH$ of a process in a system to act as a spontaneity condition are to be stated.

Concept introduction:

The spontaneous process can be defined as the process in which the free energy is released and the system moves to an energy state which is thermodynamically more stable.

Answer to Problem 4.1E

The spontaneity condition in terms of entropy is given at the constant internal energy and constant volume by the expression,

${\left(dS\right)}_{U,V}>0$

At constant volume and constant entropy, the condition for spontaneity is given in terms of internal energy as,

${\left(dU\right)}_{V,S}\le 0$

At constant pressure and entropy, the condition for spontaneity is given in terms of enthalpy as,

${\left(dH\right)}_{P,S}\le 0$

Explanation of Solution

The derivation of the spontaneity condition $\Delta S>0$ is applicable to the systems in which no pressure-volume work is done and the processes are adiabatic, that is, both $q$ and $w$ is zero. The first law of thermodynamics is $dU=dq-pdV$.

Thus, according to the first law of thermodynamics $\Delta U$ is also zero.

Thus, $\Delta S>0$ is valid when $U$ and $V$ are held constant.

The infinitesimal change in entropy is given by the expression,

${\left(dS\right)}_{U,V}>0$

According to the Clausius theorem, the entropy change is given by the expression,

$\frac{dq}{T}\le dS$

This expression can be rewritten using the $dq=dU+pdV$ as shown below.

$\frac{dU+pdV}{T}-dS\le 0$

$dU+pdV-TdS\le 0$ …(1)

At constant volume and constant entropy, the condition for spontaneity is given in terms of internal energy as,

${\left(dU\right)}_{V,S}\le 0$

For enthalpy the expression, $dH=dU+d\left(pV\right)$ can be substituted for the value of $dU$ in the equation (1) as shown below.

$\begin{array}{c}dH-d\left(pV\right)+pdV-TdS\le 0\\ dH-pdV-Vdp+pdV-TdS\le 0\end{array}$

$dH-Vdp-TdS\le 0$ …(2)

At constant pressure and entropy, the condition for spontaneity is given in terms of enthalpy as,

${\left(dH\right)}_{P,S}\le 0$

Conclusion

The spontaneity condition in terms of entropy is given at the constant internal energy and constant volume by the expression,

${\left(dS\right)}_{U,V}>0$

At constant volume and constant entropy, the condition for spontaneity is given in terms of internal energy as,

${\left(dU\right)}_{V,S}\le 0$

At constant pressure and entropy, the condition for spontaneity is given in terms of enthalpy as,

${\left(dH\right)}_{P,S}\le 0$