Question

For each of the following, specify which has more entropy per mole: a. Chlorine liquid, \(\mathrm{Cl}_{2}(\ell)\), or chlorine gas, \(\mathrm{Cl}_{2}(\mathrm{~g})\) ? b. Methane, \(\mathrm{CH}_{4}(\mathrm{~g})\), or propane, \(\mathrm{C}_{3} \mathrm{H}_{8}(\mathrm{~g})\) ? c. Sodium chloride solution, \(\mathrm{NaCl}(\mathrm{aq})\), or sodium chloride solid, \(\mathrm{NaCl}(\mathrm{s})\) ?

Short answer

1. Chlorine gas has more entropy. 2. Propane has more entropy. 3. Sodium chloride solution has more entropy.

Step-by-step solution

Understanding Entropy

Entropy is a measure of the disorder or randomness of a system. Systems with higher randomness generally have higher entropy. For gases, the molecules are more spread out and have more freedom of motion compared to liquids or solids, leading to higher entropy. For mixtures compared to pure substances, the former generally increases the randomness and hence the entropy.

Comparing Chlorine States

For chlorine (\( \ ext{Cl}_{2} \)) in liquid form versus gas, chlorine gas will have more entropy. Gas molecules are spread out and move more freely than those in liquids, increasing the disorder.

Comparing Methane and Propane

Methane (\( \ ext{CH}_{4}( ext{g}) \)) versus propane (\( \ ext{C}_{3} ext{H}_{8}( ext{g}) \)) involves comparing molecular complexity. Propane, with more atoms and bonds, offers more possible microstates and therefore has more entropy than methane.

Comparing Sodium Chloride States

Sodium chloride in aqueous solution (\( \ ext{NaCl}( ext{aq}) \)) compared to solid (\( \ ext{NaCl}( ext{s}) \)) indicates that the solution has more entropy. Dissolving NaCl increases molecular disorder as ions move freely compared to the orderly lattice of the solid.

Key concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat, work, temperature, and energy within systems. Working through the lens of entropy, thermodynamics underscores how energy is distributed or transformed in different states, such as solid, liquid, and gas. Entropy itself is a concept intertwined with the second law of thermodynamics, indicating the direction of spontaneous processes and the limitation of available energy for work. The greater the entropy of a system, the more disorder or randomness there is within it. In practical terms, entropy helps us understand phenomena such as why ice melts in a warm room or why gases expand to fill their containers. Through thermodynamic principles, we see that systems will naturally progress towards a state of equilibrium where entropy is maximized.

Molecular States

The state of molecules in a substance greatly impacts its entropy. Gases, liquids, and solids represent different molecular arrangements, each with unique properties. Gas molecules move freely and occupy a much larger volume compared to liquids and solids. This freedom results in higher entropy due to the disorder present in the system. In liquids, molecular motion is limited compared to gases, resulting in lower entropy but more than solids. Solids, with their tightly packed molecular structures in a lattice, exhibit the least entropy.

For example, comparing chlorine in its liquid and gaseous states, we can see that the gas has higher entropy. The molecules in chlorine gas have more freedom to move and are less ordered compared to the relatively confined state in the liquid.

Chemical Solutions

Chemical solutions can also significantly impact entropy. When a solid dissolves, such as sodium chloride in water, it breaks into its constituent ions, increasing the disorder of the solution compared to the crystalline structure of the solid. This transition from an ordered solid into a more random aqueous solution demonstrates an increase in entropy.

In a solution, molecules and ions are free to move around in the solvent, enhancing random motion and mixing, which increases entropy. This is why sodium chloride in aqueous form has higher entropy than its solid counterpart—because the ions are free to move about, resulting in greater molecular randomness and disorder.

• Solid: Limited movement, orderly structure
• Solution: Free movement, higher disorder

Gas Laws

Gas laws describe the behavior of gases in terms of pressure, volume, temperature, and the amount of gas present. Understanding these relationships helps explain the properties of gases and their entropy. When comparing gases to liquids or solids, gases inherently have higher entropy due to increased molecular motion and separation.

According to gas laws, such as Boyle's Law, Charles's Law, and Avogadro's Law, gases expand to occupy available space and their pressure and temperature changes affect their volume. This freedom of expansion results in greater disorder and higher entropy. For instance, let's look at methane and propane in their gaseous states. Propane having more mass and complexity than methane, also has more entropy. This is because it has a larger number of microstates due to it having more atoms and bonds, which translates to more ways to arrange the internal energy.