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Chapter 19: Problem 29

How would each of the following changes affect the number of microstates available to a system: (a) increase in temperature, (b) decrease in volume, (c) change of state from liquid to gas?

Short Answer

Expert verified
In summary, the effects of the given changes on the number of microstates in a system are: (a) increasing temperature increases the number of microstates due to increased particle movement and interactions; (b) decreasing volume decreases the number of microstates due to particle confinement and fewer possible positions; and (c) changing the state from liquid to gas increases the number of microstates due to increased particle freedom and dispersion.

Step by step solution

01

Understanding Microstates

A microstate is a specific arrangement of particles, energy, and other properties within a system at the molecular or atomic level. A system with more microstates has a higher entropy, meaning that the system is more disordered and has more possible configurations or arrangements.
02

(a) Effect of Increasing Temperature on Microstates

When the temperature of a system increases, the average kinetic energy of the particles in the system also increases. This allows the particles to have more possible locations and interactions, as they can move more freely and at higher speeds. As a result, more microstates become available to the system. In summary, an increase in temperature leads to an increase in the number of microstates.
03

(b) Effect of Decreasing Volume on Microstates

When the volume of a system decreases, the particles within the system become more confined to a smaller space. This confinement restricts the possible positions and movements of the particles. Consequently, the number of possible arrangements and interactions decrease, and the overall microstates available to the system decrease as well. In summary, decreasing the volume leads to a decrease in the number of microstates.
04

(c) Effect of Change of State from Liquid to Gas on Microstates

Changing the state of a system from liquid to gas involves an increase in temperature (and/or a decrease in pressure), leading to an increase in the average kinetic energy of the particles. In the gaseous state, the particles are more dispersed and have more freedom to move around compared to the liquid state. This increased freedom and dispersion of particles allow for more possible configurations and arrangements of the system. Consequently, a change of state from liquid to gas leads to an increase in the number of microstates. So, the changes affect the number of microstates in a system as follows: (a) increase in temperature: increase in microstates, (b) decrease in volume: decrease in microstates, (c) change of state from liquid to gas: increase in microstates.

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Most popular questions from this chapter

Calculate \(\Delta S^{\circ}\) values for the following reactions by using tabulated \(S^{\circ}\) values from Appendix \(C\). In each case explain the sign of \(\Delta S^{\circ}\). (a) \(\mathrm{N}_{2} \mathrm{H}_{4}(\mathrm{~g})+\mathrm{H}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{NH}_{3}(g)\) (b) \(\mathrm{K}(s)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{KO}_{2}(s)\) (c) \(\mathrm{Mg}(\mathrm{OH})_{2}(s)+2 \mathrm{HCl}(\mathrm{g}) \longrightarrow \mathrm{MgCl}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l)\) (d) \(\mathrm{CO}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{CH}_{3} \mathrm{OH}(g)\)

For each of the following processes, indicate whether the signs of \(\Delta S\) and \(\Delta H\) are expected to be positive, negative, or about zero. (a) A solid sublimes. (b) The temperature of a sample of \(\mathrm{Co}(s)\) is lowered from \(60^{\circ} \mathrm{C}\) to \(25^{\circ} \mathrm{C}\) (c) Ethyl alcohol evaporates from a beaker. (d) \(\mathrm{A}\) diatomic molecule dissociates into atoms. (e) A piece of charcoal is combusted to form \(\mathrm{CO}_{2}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(g)\).

One way to derive Equation \(19.3\) depends on the observation that at constant \(T\) the number of ways, \(W\), of arranging \(m\) ideal-gas particles in a volume \(V\) is proportional to the volume raised to the \(m\) power: \(W \propto V^{m}\) Use this relationship and Boltzmann's relationship between entropy and number of arrangements (Equation 19.5) to derive the equation for the entropy change for the isothermal expansion or compression of \(n\) moles of an ideal gas.

Consider a system consisting of an ice cube. (a) Under what conditions can the ice cube melt reversibly? (b) If the ice cube melts reversibly, is \(\Delta E\) zero for the process? Explain.

Consider a reaction \(\mathrm{A}_{2}(g)+\mathrm{B}_{2}(g) \rightleftharpoons 2 \mathrm{AB}(g)\), with atoms of A shown in red and atoms of B shown in blue. (a) If \(K_{c}=1\), which system is at equilibrium? (b) What is the sign of \(\Delta G\) for any process in which the contents of a reaction vessel move to equilibrium? (c) Rank the boxes in order of increasing magnitude of \(\Delta G\) for the reaction. [Sections \(19.5\) and \(19.7]\)
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