Entropy is one of the important concepts in thermodynamics and chemistry. It is a measure of the randomness or disorder of a system. The more disordered a system is, the higher its entropy. In this article, we will explore how the change in entropy is related to the change in the number of moles of a substance or a mixture.
Entropy and Molecular Motion
According to the statistical definition of entropy, it is related to the number of possible microstates of a system. A microstate is a specific configuration of the positions and velocities of the molecules in a system. The more microstates a system has, the more ways it can arrange itself, and the higher its entropy.
One factor that affects the number of microstates of a system is the molecular motion. The atoms, molecules, or ions that compose a system can undergo several types of molecular motion, such as translation, rotation, and vibration. The greater the molecular motion of a system, the greater the number of possible microstates and the higher the entropy.
The molecular motion of a system depends on its temperature and phase. At higher temperatures, the molecules have more kinetic energy and can move faster and more randomly. Therefore, higher temperatures increase the entropy of a system. Similarly, gases have more molecular motion than liquids or solids, because they have more space to move around and less intermolecular forces to restrict them. Therefore, gases have higher entropy than liquids or solids.
Entropy and Number of Moles
Another factor that affects the number of microstates of a system is the number of moles of a substance or a mixture. The more moles a system has, the more molecules it contains, and the more ways it can arrange itself. Therefore, higher number of moles increase the entropy of a system.
This effect is especially noticeable for gases, because they have more molecular motion than liquids or solids. Therefore, if the reaction involves only gases, the change in entropy is related to the total number of moles on either side of the reaction. A decrease in the number of moles on the product side means lower entropy. An increase in the number of moles on the product side means higher entropy.
For example, consider the following reaction:
2��2(�)→�2�4(�)2NO2(g)→N2O4(g)
In this reaction, two moles of gas react to form one mole of gas. Therefore, there is a decrease in the number of moles on the product side, which means lower entropy. The reaction is said to be entropy-driven.
On the other hand, consider this reaction:
�2(�)+3�2(�)→2��3(�)N2(g)+3H2(g)→2NH3(g)
In this reaction, four moles of gas react to form two moles of gas. Therefore, there is also a decrease in the number of moles on the product side, which means lower entropy. However, this reaction is not entropy-driven, because it requires an input of energy to overcome the activation barrier. The reaction is said to be enthalpy-driven.
Entropy and Mixing
Another situation where the change in entropy is related to the change in the number of moles is when two substances are mixed together. When two substances are mixed together, they tend to spread out and occupy more space than when they are separate. This increases their molecular motion and their number of possible microstates. Therefore, mixing increases the entropy of a system.
The change in entropy due to mixing can be calculated using this equation:
Δ����=−��(��ln��+��ln��)ΔmixS=−nR(χAlnχA+χBlnχB)
where �n is the total number of moles in the mixture, �R is the gas constant, ��χA and ��χB are the mole fractions of substance A and B respectively.
This equation shows that the change in entropy due to mixing depends on the total number of moles and the mole fractions of the substances. The higher the total number of moles, the higher the change in entropy. The closer the mole fractions are to 0.5, the higher the change in entropy.
For example, consider mixing 1 mole of helium gas with 1 mole of neon gas. The total number of moles is 2, and the mole fractions are both 0.5. The change in entropy due to mixing is:
Δ����=−2�(0.5ln0.5+0.5ln0.5)ΔmixS=−2R(0.5ln0.5+0.5ln0.5)
Δ����=−2�(−0.693)ΔmixS=−2R(−0.693)
Δ����=1.386�ΔmixS=1.386R
This is a positive value, which means the entropy of the system increases due to mixing.
Conclusion
In summary, the change in entropy is related to the change in the number of moles of a substance or a mixture. The more moles a system has, the more ways it can arrange itself, and the higher its entropy. This effect is especially noticeable for gases, because they have more molecular motion than liquids or solids. Therefore, if the reaction involves only gases, the change in entropy is related to the total number of moles on either side of the reaction. A decrease in the number of moles on the product side means lower entropy. An increase in the number of moles on the product side means higher entropy. Another situation where the change in entropy is related to the change in the number of moles is when two substances are mixed together. Mixing increases the entropy of a system, because it increases their molecular motion and their number of possible microstates. The change in entropy due to mixing depends on the total number of moles and the mole fractions of the substances. The higher the total number of moles, the higher the change in entropy. The closer the mole fractions are to 0.5, the higher the change in entropy.
