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Chapter 19: Problem 34

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

Expert verified
(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: 螖S = 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: 螖S = 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: 螖S_mix = -n鈧丷 * ln(x鈧�) - n鈧俁 * ln(x鈧�), with 螖S_mix being positive, indicating an increase in entropy.

Step by step solution

01

(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: 螖S = 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.
02

(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: 螖S = 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.
03

(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: 螖S_mix = -n鈧丷 * ln(x鈧�) - n鈧俁 * ln(x鈧�) where n鈧� and n鈧� are the amounts of substance 1 and 2, respectively, x鈧� and x鈧� are their respective mole fractions, and R is the gas constant. The values of ln(x鈧�) and ln(x鈧�) will be negative since mole fractions are always between 0 and 1, making 螖S_mix positive. This indicates an increase in entropy upon mixing ethanol and water.

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Most popular questions from this chapter

About \(86 \%\) of the world's electrical energy is produced by using steam turbines, a form of heat engine. In his analysis of an ideal heat engine, Sadi Carnot concluded that the maximum possible efficiency is defined by the total work that could be done by the engine, divided by the quantity of heat available to do the work (for example from hot steam produced by combustion of a fuel such as coal or methane). This efficiency is given by the ratio \(\left(T_{\text {high }}-T_{\text {low }}\right) / T_{\text {high }}\), where \(T_{\text {high }}\) is the temperature of the heat going into the engine and \(T_{\text {low }}\) is that of the heat leaving the engine. (a) What is the maximum possible efficiency of a heat engine operating between an input temperature of \(700 \mathrm{~K}\) and an exit temperature of \(288 \mathrm{~K} ?(\mathrm{~b})\) Why is it important that electrical power plants be located near bodies of relatively cool water? (c) Under what conditions could a heat engine operate at or near \(100 \%\) efficiency? (d) It is often said that if the energy of combustion of a fuel such as methane were captured in an electrical fuel cell instead of by burning the fuel in a heat engine, a greater fraction of the energy could be put to useful work. Make a qualitative drawing like that in Figure \(5.10\) that illustrates the fact that in principle the fuel cell route will produce more useful work than the heat engine route from combustion of methane.

The volume of \(0.100 \mathrm{~mol}\) of helium gas at \(27^{\circ} \mathrm{C}\) is increased isothermally from \(2.00 \mathrm{~L}\) to \(5.00 \mathrm{~L}\). Assuming the gas to be ideal, calculate the entropy change for the process.

Predict the sign of \(\Delta S_{\text {sys }}\) for each of the following processes: (a) Gaseous Ar is liquefied at \(80 \mathrm{~K}\). (b) Gaseous \(\mathrm{N}_{2} \mathrm{O}_{4}\) dissociates to form gaseous \(\mathrm{NO}_{2}\). (c) Solid potassium reacts with gaseous \(\mathrm{O}_{2}\) to form solid potassium superoxide, \(\mathrm{KO}_{2}\). (d) Lead bromide precipitates upon mixing \(\mathrm{Pb}\left(\mathrm{NO}_{3}\right)_{2}(a q)\) and \(\mathrm{KBr}(a q)\)

Consider the vaporization of liquid water to steam at a pressure of 1 atm. (a) Is this process endothermic or exothermic? (b) In what temperature range is it a spontaneous process? (c) In what temperature range is it a nonspontaneous process? (d) At what temperature are the two phases in equilibrium?

(a) What is meant by calling a process irreversible? (b) After an irreversible process the system is restored to its original state. What can be said about the condition of the surroundings after the system is restored to its original state? (c) Under what conditions will the condensation of a liquid be an irreversible process?
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