Question

Consider a process in which an ideal gas changes from state 1 to state 2 in such a way that its temperature changes from \(300 \mathrm{~K}\) to \(200 \mathrm{~K}\). (a) Does the temperature change depend on whether the process is reversible or irreversible? (b) Is this process isothermal? (c) Does the change in the internal energy, \(\Delta E\), depend on the particular pathway taken to carry out this change of state?

Short answer

(a) No, the temperature change does not depend on whether the process is reversible or irreversible. (b) No, this process is not isothermal. (c) No, the change in internal energy does not depend on the particular pathway taken to carry out the change of state.

Step-by-step solution

Determine if the temperature change depends on nature of process

Temperature is a state variable, and the change in temperature from state 1 to state 2 (∆T) depends only on the initial and final states of the system, not on the nature of the process (reversible or irreversible). Hence, the temperature change does not depend on whether the process is reversible or irreversible. (a) Answer: No, the temperature change does not depend on whether the process is reversible or irreversible.

Determine if the process is isothermal

An isothermal process involves a change in the system while maintaining a constant temperature. In this case, the initial temperature is 300 K, and the final temperature is 200 K. Since the temperature changes, this process is not isothermal. (b) Answer: No, this process is not isothermal.

Determine if the change in internal energy depends on the pathway taken

In an ideal gas, the change in internal energy (∆E) depends only on the change in temperature and not on the pathway taken. This is because internal energy is a state function that depends on the initial and final states of the system, not on the process itself. Mathematically, the internal energy of an ideal gas can be expressed as: \(\Delta E = nC_v\Delta T\) Where: - n represents the number of moles of the gas - \(C_v\) represents the molar specific heat at constant volume - ΔT represents the change in temperature between state 1 and state 2. Since the internal energy depends only on the change in temperature, which is independent of the pathway taken, the internal energy change (∆E) is also independent of the particular pathway taken to carry out the change of state. (c) Answer: No, the change in internal energy does not depend on the particular pathway taken to carry out the change of state.

Key concepts

State Function

When we talk about state functions, we're referring to properties of a system that depend solely on the current state of the system, not how it got there. Temperature, pressure, and volume are examples of state functions. This means that no matter what path or process the system follows, a state function like temperature will only look at where the system starts and ends.

In the context of our ideal gas example, the temperature change from 300 K to 200 K is determined by these two states alone. It doesn’t matter whether the gas changes temperature through a slow, reversible path or a quick, irreversible one. The change remains the same. This is a fundamental property of state functions—they ignore the specifics of the process and focus only on the beginning and end points.

• State functions are independent of the process path.
• Temperature is a state function.
• Temperature change is based only on initial and final states.

Internal Energy

The concept of internal energy in thermodynamics is all about the energy contained within a system. For an ideal gas, internal energy is closely tied to temperature. The change in internal energy, represented as \(\Delta E\), is computed based on changes in temperature, regardless of the path taken to reach that temperature difference.

Mathematically, this can be expressed as: \[\Delta E = nC_v\Delta T\]Here, \(n\) is the number of moles of gas, \(C_v\) is the molar specific heat at constant volume, and \(\Delta T\) is the change in temperature between the initial and final states. This equation highlights that \(\Delta E\) depends solely on temperature change, reinforcing that internal energy is a state function.

• Internal energy for ideal gases is linked to temperature.
• The formula \(\Delta E = nC_v\Delta T\) shows dependence on temperature change.
• Changes in internal energy are the same, no matter the process path.

Temperature Change

In thermodynamics, temperature is a crucial measurement as it influences many other properties and processes in a system. A temperature change indicates energy being added or removed from a system. For ideal gases, this alteration is an indicator of change in internal energy.

A temperature change from 300 K to 200 K, as in our example, results from either absorbing or releasing heat. However, this change is strictly independent of the way the gas transitions from its initial to its final state. Whether the process was a smooth, reversible interaction or a rapid, irreversible transformation, the temperature change remains consistent.

• Temperature is a direct measure of thermal energy.
• Temperature change affects internal energy.
• Independence from process type: reversible or irreversible.