Question

Which substance has the higher entropy? (a) a sample of pure silicon (to be used in a computer chip) or a piece of silicon containing a trace of another element such as boron or phosphorus (b) \(\mathrm{O}_{2}(\mathrm{g})\) at \(0^{\circ} \mathrm{C}\) or \(\mathrm{O}_{2}(\mathrm{g})\) at \(-50^{\circ} \mathrm{C}\) (c) \(\mathrm{I}_{2}(\mathrm{s})\) or \(\mathrm{I}_{2}(\mathrm{g}),\) both at room temperature (d) one mole of \(\mathrm{O}_{2}(\mathrm{g})\) at 1 bar pressure or one mole of \(\mathrm{O}_{2}(\mathrm{g})\) at 0.01 bar pressure (both at \(298 \mathrm{K})\)

Short answer

Higher entropy: (a) Silicon with impurity, (b) O2 at 0°C, (c) I2(gas), (d) O2 at 0.01 bar.

Step-by-step solution

Understand Entropy

Entropy is a measure of disorder or randomness in a system. Higher entropy means more disorder. Factors that affect entropy include temperature, phase of matter, and the purity or composition of a substance.

Analyze Case (a)

In part (a), we compare pure silicon to silicon with a trace of another element. Adding another element, such as boron or phosphorus, introduces more disorder in the silicon lattice, thus increasing entropy. Therefore, silicon with impurities has higher entropy than pure silicon.

Analyze Case (b)

In part (b), we compare oxygen gas at two temperatures: 0°C and -50°C. Higher temperatures cause molecules to move more freely, increasing disorder. Thus, \(\mathrm{O}_{2}(\mathrm{g})\) at 0°C has higher entropy than at -50°C.

Analyze Case (c)

In part (c), we compare iodine in solid form to gaseous form at room temperature. The gas phase has higher entropy than the solid phase due to more molecular movement and space. Therefore, \(\mathrm{I}_{2}(\mathrm{g})\) has higher entropy than \(\mathrm{I}_{2}(\mathrm{s})\).

Analyze Case (d)

In part (d), we're comparing pressures of oxygen gas. Lower pressure means the molecules are further apart, providing more space for movement and thus more disorder. Therefore, \(\mathrm{O}_{2}(\mathrm{g})\) at 0.01 bar has higher entropy than at 1 bar.

Key concepts

Chemical Thermodynamics

Chemical thermodynamics is a branch of physical chemistry focused on the study of energy transformations in chemical processes. One of its fundamental concepts is entropy, a measure of disorder or randomness in a system. Entropy is crucial because it helps us understand how energy disperses and becomes more spread out in a system. In chemical reactions, reactions tend to proceed in the direction that increases the total entropy of the system and its surroundings. This is encapsulated in the Second Law of Thermodynamics, which states that in any spontaneous process, the total entropy of a system and its environment always increases.

Another important aspect of chemical thermodynamics is the concept of Gibbs free energy. This is the energy associated with a chemical reaction that can be used to do work. Gibbs free energy takes into account both enthalpy and entropy, giving a more comprehensive picture of the energy dynamics in a chemical process. The relationship between these entities is given by the equation: \[ G = H - TS \] where \( G \) is the Gibbs free energy, \( H \) is the enthalpy, \( T \) is the temperature, and \( S \) is the entropy.

• Entropy is a key concept in chemical thermodynamics.
• Spontaneous processes increase the total entropy of a system and its surroundings.
• Gibbs free energy helps predict the feasibility of chemical reactions.

Phase Transition

Phase transition refers to the transformation from one state of matter to another, such as from solid to liquid, liquid to gas, or vice versa. These transitions occur because of changes in temperature or pressure and are accompanied by changes in entropy. During a phase transition, the energy of the system changes as the molecular motion also changes.

For example, when a solid melts into a liquid, the orderly arrangement of molecules in the solid becomes more random, resulting in an increase in entropy. Similarly, when a liquid vaporizes into a gas, there's an even greater increase in entropy because the molecules are far apart and move freely. In contrast, freezing and condensation represent decreases in entropy as the system becomes more ordered.

• Melting and vaporization increase entropy due to less molecular ordering.
• Freezing and condensation decrease entropy as ordering increases.

Understanding phase transitions is crucial in fields such as materials science, meteorology, and chemical engineering, where controlling the properties and phases of materials is essential.

Molecular Motion

Molecular motion encompasses the movement of molecules within different states of matter. It plays a critical role in determining the entropy of a system. In solids, molecules are tightly packed and vibrate around fixed positions, resulting in low entropy. In liquids, molecules have more freedom and can slide past one another, leading to higher entropy compared to solids. In gases, molecules move rapidly and independently, which corresponds to the highest level of entropy.

Temperature is a key factor influencing molecular motion. Higher temperatures provide molecules with more energy, causing them to move more energetically. This increased motion raises the entropy of the system. For example, in gases, heating leads to an increase in the speed and range of molecular movement, which disperses energy and increases disorder.

• Temperature impacts the speed and freedom of molecular motion.
• More molecular freedom correlates with higher entropy.

The study of molecular motion is vital in understanding reaction rates, diffusion, and the physical properties of substances across different states.

Impurities in Solids

Impurities in solids refer to the presence of foreign atoms or molecules within a crystalline lattice. These impurities disrupt the orderly arrangement of the atoms, leading to an increase in the system's entropy. As impurities introduce irregularities and disorder within the solid, they provide additional states in which the system can exist.

For example, adding an impurity such as boron or phosphorus to silicon, used in semiconductors, increases the entropy of the silicon crystal. These elements fit into the crystal lattice differently than silicon atoms, creating a less ordered structure and thus more disorder.

• Impurities create disorder by introducing atomic irregularities.
• Higher entropy results from increased disorder in the crystal lattice.

Understanding the role of impurities is crucial in materials science and engineering, as they can significantly alter the mechanical, thermal, and electrical properties of materials.