Problem 34 - Chemistry: The Central Science

Question

How does the entropy of the system change when (a) the temperature of the system increases, (b) the volume of a gas increases, (c) equal volumes of ethanol and water are mixed to form a solution.

Step-by-Step Solution

Answer

(a) When the temperature of the system increases, the change in entropy (∆S) is positive, indicating an increase in disorder. This is due to the formula ∆S = nC ln(T2/T1), where T2 > T1. (b) When the volume of a gas increases, the change in entropy (∆S) is also positive, indicating an increase in disorder. This is calculated using the formula ∆S = nR ln(V2/V1), where V2 > V1. (c) When equal volumes of ethanol and water are mixed, the change in entropy (∆S) is positive, indicating increased disorder in the system. This is because the total entropy of the mixed solution (S_mix) is greater than the sum of the separate entropies of ethanol and water.

Step 1: (a) Change in entropy when the temperature of the system increases

To find the change in entropy when the temperature of the system increases, we use the formula: ∆S = nC ln(T2/T1) where n is the number of moles of the substance, C is the molar heat capacity at constant volume or pressure (depending on the conditions given), and T1 and T2 are the initial and final temperatures of the system, respectively. In this case, since the temperature of the system increases, T2 > T1, so the ratio T2/T1 > 1. Therefore, the natural logarithm ln(T2/T1) is positive, which means the change in entropy ∆S is positive. A positive change in entropy indicates that the disorder of the system has increased as the temperature increased.

Step 2: (b) Change in entropy when the volume of a gas increases

To find the change in entropy when the volume of a gas increases, we use the formula: ∆S = nR ln(V2/V1) where n is the number of moles of the gas, R is the universal gas constant, and V1 and V2 are the initial and final volumes of the gas, respectively. In this case, the volume of the gas increases, so V2 > V1, and the ratio V2/V1 > 1. Thus, the natural logarithm ln(V2/V1) is positive, which means the change in entropy ∆S is also positive. A positive change in entropy indicates that the disorder of the gas has increased as the volume increased.

Step 3: (c) Change in entropy when equal volumes of ethanol and water are mixed to form a solution

When two substances, such as ethanol and water, are mixed, the total entropy of the system increases due to the increased number of possible microstates. This is because when ethanol and water molecules mix, there is a greater number of possible arrangements for the molecules compared to the entropy of the separate substances. Before mixing, there are a certain number of microstates available for ethanol (let's call this S_ethanol) and a certain number of microstates available for water (S_water). After mixing, the number of microstates available for the solution (S_mix) is greater than the sum of the microstates for the separate substances. Thus, the change in entropy ∆S for this case can be calculated as: ∆S = S_mix - (S_ethanol + S_water) Since the total entropy after mixing is greater than the sum of the entropies of the separate substances, the change in entropy ∆S is positive. A positive change in entropy indicates that the disorder of the system has increased upon mixing equal volumes of ethanol and water.

Key Concepts

Entropy and Temperature

Entropy, a fundamental concept in thermodynamics, is often associated with the level of disorder within a given system. With respect to temperature, entropy changes can be quite intuitive: as temperature increases, so does the kinetic energy of the particles within a substance. These particles move more vigorously, leading to a greater number of accessible microstates and, as a result, an increase in the system's entropy.

The relationship is quantified by the equation \( \text{\text{Δ}}S = nC \text{\text{ln}}(\text{\text{T}}2/\text{\text{T}}1) \), where 'n' represents the number of moles, 'C' is the molar heat capacity, and 'T1' and 'T2' are initial and final temperatures, respectively. Since an increase in temperature (\text{\text{T}}2 > \text{\text{T}}1) yields a positive logarithm, the entropy change \text{\text{Δ}}S is positive, reflecting increased disorder.

Entropy and Gas Volume

When considering how entropy is influenced by the volume of a gas, Boyle's Law is not directly related, yet it offers insight into the behavior of gases. As gas expands into a larger volume (V2 > V1), it exhibits more spatial freedom, which translates into a higher number of microstates and thus an increase in entropy.

The equation \( \text{\text{Δ}}S = nR \text{\text{ln}}(\text{\text{V}}2/\text{\text{V}}1) \) explains this relationship, where 'R' is the universal gas constant. A greater volume results in a positive value of \text{\text{Δ}}S, indicating more disorder. Gases inherently have high entropy due to their molecular freedom, and this is enhanced as volume increases.

Mixing Entropy

Mixing entropy is a fascinating aspect of thermodynamics that describes the increase in entropy when two or more different substances are combined. Intuitively, this can be thought of as adding to the disorder of a system due to the myriad ways the different molecules can arrange themselves when mixed.

When substances like ethanol and water mix, the distinct molecular interactions between different types of molecules create a much greater number of possible configurations (microstates). Consequently, the system's disorder, as measured by entropy, escalates. The calculation \( \text{\text{Δ}}S = S_{\text{\text{mix}}} - (S_{\text{\text{ethanol}}} + S_{\text{\text{water}}}) \) enumerates this change, with \text{\text{Δ}}S being positive, which confirms an entropy increase due to mixing.

Thermodynamics

Thermodynamics is a branch of physics concerned with heat and temperature and their relation to energy and work. It encompasses the principles that govern the transformation and conservation of energy within systems. Entropy is a central theme in thermodynamics, serving as a measure of the number of ways in which a system can be arranged, often interpreted as a measure of disorder.

Four fundamental laws dictate thermodynamic processes, starting from the zeroth law, which establishes temperature, through to the first, second, and third laws that pronounce on energy conservation, entropy, and the conditions of absolute zero, respectively. Whether examining heat engines, refrigerators, or the universe itself, the laws of thermodynamics and the concept of entropy provide the backbone of understanding how energy is transferred and how systems evolve towards equilibrium.