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Chapter 19: Problem 21

(a) How can we calculate \(\Delta S\) foran isothermal process? (b) Does \(\Delta S\) for a process depend on the path taken from the initial to the final state of the system? Explain.

Short Answer

Expert verified
(a) To calculate 螖S for an isothermal process, we use the formula 螖S = \( \int_{i}^{f} \frac{dQ_{rev}}{T} \), where 螖S is the change in entropy, dQ_rev is the infinitesimal amount of reversible heat, and T is the constant temperature. This formula is valid only for reversible processes. (b) 螖S for a process does not depend on the path taken from the initial to the final state of the system because entropy is a state function, determined only by the current state of the system and not the path taken to reach that state.

Step by step solution

01

(a) Calculating the Change in Entropy (螖S) for an Isothermal Process

: To calculate the change in entropy (螖S) for an isothermal process, we can use the following formula: 螖S = \( \int_{i}^{f} \frac{dQ_{rev}}{T} \) where 螖S is the change in entropy, dQ_rev is the infinitesimal amount of reversible heat and T is the constant temperature of the process. Note that the integration goes from the initial state (i) to the final state (f). It is important to remember that this formula is only valid for a reversible process. If the process is irreversible, the entropy change of the system can still be calculated by first finding a reversible path between the same initial and final states and then using the formula above.
02

(b) Dependency of 螖S on the Path Taken from Initial to Final State

: The change in entropy (螖S) for a process depends only on the initial and final states of the system, and not on the path taken between them. This is because entropy is a state function, meaning its value is determined by the current state of the system and not by the path taken to reach that state. To understand this, we should take a closer look at the formula for calculating entropy change: 螖S = \( \int_{i}^{f} \frac{dQ_{rev}}{T} \) Notice that the integral depends only on the initial (i) and final (f) states of the system and the temperature (T). There is no term for the specific path taken. Therefore, the change in entropy (螖S) for a process does not depend on the path taken from the initial to the final state of the system.

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Most popular questions from this chapter

For each of the following pairs, indicate which substance possesses the larger standard entropy: (a) \(1 \mathrm{~mol}\) of \(\mathrm{P}_{4}(\mathrm{~g})\) at \(300{ }^{\circ} \mathrm{C}, 0.01 \mathrm{~atm}\), or \(1 \mathrm{~mol}\) of \(\mathrm{As}_{4}(\mathrm{~g})\) at \(300{ }^{\circ} \mathrm{C}, 0.01 \mathrm{~atm}\); (b) \(1 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}(g)\) at \(100^{\circ} \mathrm{C}, 1 \mathrm{~atm}\), or \(1 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}(\mathrm{l})\) at \(100^{\circ} \mathrm{C}, 1 \mathrm{~atm} ;\) (c) \(0.5 \mathrm{~mol}\) of \(\mathrm{N}_{2}(g)\) at \(298 \mathrm{~K}, 20\) - \(\mathrm{L}\) volume, or \(0.5 \mathrm{~mol} \mathrm{CH}_{4}(g)\) at \(298 \mathrm{~K}, 20-\mathrm{L}\) volume; (d) \(100 \mathrm{~g}\) \(\mathrm{Na}_{2} \mathrm{SO}_{4}(s)\) at \(30^{\circ} \mathrm{C}\) or \(100 \mathrm{~g} \mathrm{Na}_{2} \mathrm{SO}_{4}(a q)\) at \(30^{\circ} \mathrm{C} .\)

(a) Give two examples of endothermic processes that are spontaneous. (b) Give an example of a process that is spontaneous at one temperature but nonspontaneous at a different temperature.

The normal freezing point of 1 -propanol \(\left(\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}\right)\) is \(-127{ }^{\circ} \mathrm{C}\). (a) Is the freezing of 1-propanol an endothermic or exothermic process? (b) In what temperature range is the freezing of 1-propanol a spontaneous process? (c) In what temperature range is it a nonspontaneous process? (d) Is there any temperature at which liquid and solid 1-propanol are in equilibrium? Explain.

Consider the reaction \(6 \mathrm{H}_{2}(\mathrm{~g})+\mathrm{P}_{4}(g) \longrightarrow 4 \mathrm{PH}_{3}(g)\). (a) Using data from Appendix C, calculate \(\Delta G^{\circ}\) at \(298 \mathrm{~K}\). (b) Calculate \(\Delta G\) at \(298 \mathrm{~K}\) if the reaction mixture consists of \(8.0 \mathrm{~atm}\) of \(\mathrm{H}_{2}, 0.050 \mathrm{~atm}\) of \(\mathrm{P}_{4}\), and \(0.22 \mathrm{~atm}\) of \(\mathrm{PH}_{3}\).

A particular reaction is spontaneous at \(450 \mathrm{~K}\). The enthalpy change for the reaction is \(+34.5 \mathrm{~kJ} .\) What can you conclude about the sign and magnitude of \(\Delta S\) for the reaction?
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