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Chapter 19: Problem 17

Consider the following process: a system changes from state 1 (initial state) to state 2 (final state) in such a way that its temperature changes from \(300 \mathrm{~K}\) to \(400 \mathrm{~K}\). (a) Is this process isothermal? (b) Does the temperature change depend on the particular pathway taken to carry out this change of state? (c) Does the change in the internal energy, \(\Delta E\), depend on whether the process is reversible or irreversible?

Short Answer

Expert verified
(a) No, this process is not isothermal as the temperature changes from 300 K to 400 K. (b) The temperature change does not depend on the specific pathway taken. (c) The internal energy change, \(\Delta E\), does not depend on whether the process is reversible or irreversible as it is only a function of the initial and final states, not the process's nature connecting those states.

Step by step solution

01

(a) Identifying if the process is isothermal

An isothermal process is a process that occurs at a constant temperature. Here, the initial temperature is 300 K, and the final temperature is 400 K. Since the temperature changes during the process, we can conclude that it is not an isothermal process.
02

(b) Temperature change and pathway dependency

The given information tells us that the initial state of the system has a temperature of 300 K and the final state has a temperature of 400 K. The temperature change, which is the difference between final and initial temperatures, depends only on the initial and final states (\( \Delta T = T_{final} - T_{initial} = 400 K - 300 K = 100 K\)). It does not depend on the specific pathway taken to achieve this change in state. Therefore, the temperature change is independent of the particular pathway.
03

(c) Internal energy change dependency on process reversibility

The change in internal energy, denoted as \(\Delta E\), depends on the initial and final states of the system, and not on the pathway taken or the nature of the process (reversible or irreversible). Essentially, the internal energy is a state function, meaning it only depends on the initial and final states and not on the details of the process connecting those states. In conclusion, (a) No, this process is not isothermal as the temperature changes from 300 K to 400 K, (b) The temperature change does not depend on the specific pathway taken, and (c) The internal energy change, \(\Delta E\), does not depend on whether the process is reversible or irreversible as it is only a function of the initial and final states, not the process's nature connecting those states.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isothermal Process
An isothermal process is a fascinating concept in thermodynamics where the temperature of a system remains constant throughout the entire process. This means every part of the pathway from the initial state to the final state is at the same temperature.
Such processes often occur when a system is in thermal equilibrium with a heat reservoir. To better visualize, think of an ideal gas in a piston. If the gas expands or compresses but the temperature stays the same, any heat added or removed is counterbalanced by the work done by the system, keeping the net internal energy change zero.
However, if the temperature changes, as in the given exercise from 300 K to 400 K, the process cannot be classified as isothermal. This change indicates energy was either absorbed or released, altering the temperature profile of the system.
Pathway Dependency
The idea of pathway dependency deals with how changes are achieved in thermodynamic processes. For certain properties like temperature change, the pathway taken from the initial state to the final state makes no difference.
In simpler terms, if you start at state 1 with a temperature of 300 K and end at state 2 with a temperature of 400 K, the difference in temperature, denoted as \( \Delta T = 100 \) K, depends solely on these states, not on how you got there.
  • This is because temperature is a state function, a property dependent only on the current state of the system, not the process used to reach that state.
In our exercise, this means even if energy is added or subtracted through different methods or paths, the overall temperature change remains the same.
Internal Energy Change
Internal energy is a crucial concept in thermodynamics, indicating the total energy contained within a system. It's important to note that this change in internal energy, represented by \( \Delta E \), is influenced only by the initial and final states of the system.Unlike some thermodynamic properties, internal energy does not depend on whether the process that connects these states is reversible or irreversible. Instead, it is a state function, making internal energy invariant concerning the pathway taken.
For instance, when moving from a state with a certain amount of energy at 300 K to another at 400 K, the internal energy's change will rely solely on these thermal energies and not on how the change occurs.
  • This helps clarify why, regardless of the reversibility of the process, the internal energy outcome remains consistent.
In summary, internal energy focuses on state change rather than the journey, highlighting the core principle of state functions in thermodynamics.

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Most popular questions from this chapter

Predict which member of each of the following pairs has the greater standard entropy at \(25^{\circ} \mathrm{C}:\) (a) \(\mathrm{C}_{6} \mathrm{H}_{6}(l)\) or \(\mathrm{C}_{6} \mathrm{H}_{6}(g)\), (b) \(\mathrm{CO}(g)\) or \(\mathrm{CO}_{2}(g)\) (c) \(1 \mathrm{~mol} \mathrm{~N}_{2} \mathrm{O}_{4}(g)\) or \(2 \mathrm{~mol} \mathrm{NO}_{2}(g)\) (d) \(\mathrm{HCl}(g)\) or \(\mathrm{HCl}(a q) .\) Use Appendix \(\mathrm{C}\) to find the standard entropy of each substance.

Indicate whether each statement is true or false. (a) \(\Delta S\) depends on whether the process is reversible or irreversible. \((\mathbf{b})\) If a system undergoes an irreversible change, the entropy of the universe increases. (c) Only if the change in entropy of the system is exactly matched by an equal and opposite change in the entropy of the surroundings, the system undergoes a reversible process. (d) If the entropy change of the system is zero, the system undergoes a reversible process.

Consider a system that consists of two standard playing dice, with the state of the system defined by the sum of the values shown on the top faces. (a) The two arrangements of top faces shown here can be viewed as two possible microstates of the system. Explain. (b) To which state does each microstate correspond? (c) How many possible states are there for the system? (d) Which state or states have the highest entropy? Explain. (e) Which state or states have the lowest entropy? Explain. (f) Calculate the absolute entropy of the two-dice system.

Indicate whether each statement is true or false. (a) A reaction that is spontaneous in one direction will be nonspontaneous in the reverse direction under the same reaction conditions. (b) All spontaneous processes are fast. (c) Most spontaneous processes are reversible. (d) An isothermal process is one in which the system loses no heat. (e) The maximum amount of work can be accomplished by an irreversible process rather than a reversible one.

Today, most candles are made of paraffin wax. A typical component of paraffin wax is the hydrocarbon \(\mathrm{C}_{31} \mathrm{H}_{64}\) which is solid at room temperature. (a) Write a balanced equation for the combustion of \(\mathrm{C}_{31} \mathrm{H}_{64}(s)\) to form \(\mathrm{CO}_{2}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(g) .\) ( \(\mathbf{b}\) ) Without using thermochemical data, predict whether \(\Delta G^{\circ}\) for this reaction is more negative or less negative than \(\Delta H^{\circ}\).
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