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.

Short answer

(a) When the temperature of the system increases, the entropy also increases as the randomness or disorder of the particles increases. This is mathematically represented by the equation: ΔS = C_p * ln(T2 / T1), with T2 > T1. (b) When the volume of a gas increases, the entropy increases as the gas particles have more space to move around, and their randomness or disorder increases. This is mathematically described by the equation: ΔS = nR * ln(V2 / V1), with V2 > V1. (c) Mixing equal volumes of ethanol and water increases entropy due to the increased number of possible arrangements and interactions between the particles of the two substances. The entropy change due to mixing is given by the equation: ΔS_mix = -n₁R * ln(x₁) - n₂R * ln(x₂), with ΔS_mix being positive, indicating an increase in entropy.

Step-by-step solution

(a) Increase in temperature and its effect on entropy

As the temperature of the system increases, the randomness or disorder of the particles in the system will also increase, which in turn increases the entropy. In mathematical terms, the entropy change is given by: ΔS = C_p * ln(T2 / T1) where C_p is the heat capacity of the system at constant pressure, T1 is the initial temperature, and T2 is the final temperature. As T2 > T1, the natural logarithm in the equation will be positive, resulting in an increase in entropy.

(b) Increase in volume and its effect on entropy

When the volume of a gas increases, the gas particles have more space to move around. Therefore, the randomness or disorder of the particles' positions and momenta will increase, resulting in an increase in entropy. For an ideal gas, the entropy change can be described by: ΔS = nR * ln(V2 / V1) where n is the amount of gas, R is the gas constant, V1 is the initial volume, and V2 is the final volume. Since V2 > V1, the natural logarithm in the equation will be positive, resulting in an increase in entropy.

(c) Mixing ethanol and water and its effect on entropy

When two substances are mixed (such as ethanol and water), the randomness or disorder will also increase. This is due to the increased number of possible arrangements and interactions between the particles of the two substances. Mixing generally results in an increase in entropy. The entropy change due to mixing can be described by: ΔS_mix = -n₁R * ln(x₁) - n₂R * ln(x₂) where n₁ and n₂ are the amounts of substance 1 and 2, respectively, x₁ and x₂ are their respective mole fractions, and R is the gas constant. The values of ln(x₁) and ln(x₂) will be negative since mole fractions are always between 0 and 1, making ΔS_mix positive. This indicates an increase in entropy upon mixing ethanol and water.