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Chapter 9: Problem 4

What thermodynamic condition must be met for a state of equilibrium to exist?

Short Answer

Expert verified
Answer: The thermodynamic condition for a state of equilibrium to exist is that the system's total energy must be minimized, and it should not change with time. This means that the derivative of the total energy with respect to an internal parameter (such as volume or entropy) must be equal to zero, ensuring that the system is at its lowest energy state and remains stable.

Step by step solution

01

Understanding the concept of thermodynamic equilibrium

Thermodynamic equilibrium is a state where the macroscopic properties of a system, such as pressure, temperature, and volume, are uniform and don't change with time. It means that there is no net exchange of energy or matter taking place, and the system is at its lowest energy state.
02

Identifying the conditions for thermodynamic equilibrium

A system is in equilibrium if it satisfies the following three conditions: 1. Mechanical equilibrium: There is no unbalanced force in the system that can lead to a change in volume or shape. 2. Thermal equilibrium: All parts of the system are at the same temperature. 3. Chemical equilibrium: There is no net exchange of matter between different parts of the system or with the surroundings.
03

Understanding the thermodynamic condition for equilibrium

The thermodynamic condition for a state of equilibrium to exist is that the system's total energy must be minimized, and it should not change with time. Mathematically, it can be represented as: The total energy E_tot = E_int + E_ext where E_tot is the total energy of the system, E_int is the internal energy of the system, and E_ext is the external energy of the system.
04

Explanation of the thermodynamic condition

For the system to be in equilibrium, the total energy must be minimized with respect to any internal adjustments. To satisfy this condition, the derivative of the total energy with respect to an internal parameter (such as volume or entropy) must be equal to zero: \(\frac{\partial E_{tot}}{\partial x} = 0\) where x is the internal parameter. This condition essentially means that any small changes in the system's properties will not lead to a decrease in the total energy, ensuring that the system is at its lowest energy state and remains stable. It also implies that all thermodynamic processes are reversible since no energy is lost or gained during the process, keeping the total energy constant.

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Most popular questions from this chapter

Figure \(9.36\) is the tin-gold phase diagram, for which only single-phase regions are labeled. Specify temperature-composition points at which all eutectics, eutectoids, peritectics, and congruent phase transformations occur. Also, for each, write the reaction upon cooling.

Is it possible to have a magnesium-lead alloy in which the mass fractions of primary \(\alpha\) and total \(\alpha\) are \(0.60\) and \(0.85\), respectively, at \(460^{\circ} \mathrm{C}\left(860^{\circ} \mathrm{F}\right)\) ? Why or why not?

Compute the mass fractions of proeutectoid ferrite and pearlite that form in an iron-carbon alloy containing \(0.35 \mathrm{wt} \% \mathrm{C}\).

A steel alloy contains \(95.7 \mathrm{wt} \% \mathrm{Fe}, 4.0 \mathrm{wt} \% \mathrm{~W}\), and \(0.3 \mathrm{wt} \% \mathrm{C}\). (a) What is the eutectoid temperature of this alloy? (b) What is the eutectoid composition? (c) What is the proeutectoid phase? Assume that there are no changes in the positions of other phase boundaries with the addition of \(\mathrm{W}\).

Plot the mass fraction of phases present versus temperature for a \(40 \mathrm{wt} \% \mathrm{Sn}-60 \mathrm{wt} \% \mathrm{~Pb}\) alloy as it is slowly cooled from \(250^{\circ} \mathrm{C}\) to \(150^{\circ} \mathrm{C}\).
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