Question

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

(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

(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.

(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.

(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.

Key concepts

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.