Chapter 5, Problem 5.1CP

Under what conditions is $\Delta S<0$ for a spontaneous process?

Interpretation Introduction

Interpretation:The condition for which $\Delta \text{S <0}$ for a spontaneous process needs to be explained.

Concept Introduction:The Gibb’s equation of thermodynamic purposed a relation between $\text{ΔS}$ , $\text{ΔH}$ and $\text{ΔG}$ with temperature. With the help of this equation,one can predict the change in $\text{ΔS}$ , $\text{ΔH}$ and $\text{ΔG}$ . The equation is represented as follows:

  ${\Delta }_r{G}^o={\Delta }_r{H}^o-T{\Delta }_r{S}^o$

For any reaction, the $\text{ΔH}$ and $\text{ΔS}$ can be calculated with the help of:

  $\begin{array}{l}\text{ΔrS°= }\Sigma \text{ΔrS°product - }\sum \text{ΔrS°reactant}\\ \text{ΔrH°= }\Sigma \text{ΔrH°product - }\sum \text{ΔrH°reactant}\end{array}$

Answer to Problem 5.1CP

For a spontaneous reaction, $\text{ΔS}+\text{ΔSsurr}>0$ .Hence, the process will be spontaneous only if the $\text{ΔS<0}$ and ${\text{ΔS}}_{\text{surr}}\text{ > 0}$ so that ${\text{ΔS}}_{\text{surr}}\text{ > ΔS }$

Explanation of Solution

A system is the substance that is used to study in thermodynamic. The remaining part of universe is said to be surrounding. Overall one can say that universe is made from system and surrounding which are separated by an imaginary boundary. The entropy change for surrounding can be written as $\text{ΔSsurr}$ and the entropy change for system is $\text{ΔSsys}$ Hence $\text{ΔSuni}$ represents the change in entropy of the universe and equal to the sum of $\text{ΔSsurr}$ and $\text{ΔSsys}$ .

  $\text{ΔSuni}$ = $\text{ΔSsurr}$ + $\text{ΔSsys}$

According to the second law of thermodynamics, a spontaneous process causes the entropy of the universe to increase. Overall the net change in entropy for the system and the surrounding must be positive.

Hence, for a spontaneous reaction, $\text{ΔS}+\text{ΔSsurr}>0$ . Hence, the process will only spontaneous if the $\text{ΔS<0}$ and ${\text{ΔS}}_{\text{surr}}\text{ > 0}$ and ${\text{ΔS}}_{\text{surr}}\text{ > ΔS }$

Conclusion

For a spontaneous reaction, $\text{ΔS}+\text{ΔSsurr}>0$ .Hence, the process will only spontaneous if the $\text{ΔS<0}$ and ${\text{ΔS}}_{\text{surr}}\text{ > 0}$ and ${\text{ΔS}}_{\text{surr}}\text{ > ΔS }$ .