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 K to 200 K. (c) Does the change in the internal energy, ΔE, depend on the particular pathway taken to carry out this change of state?

Verified step by step guidance

1. Understand that the internal energy change, \( \Delta E \), for an ideal gas depends only on the initial and final states, not on the path taken.
2. Recall that for an ideal gas, the internal energy is a function of temperature only, specifically \( E = \frac{3}{2}nRT \) for a monatomic ideal gas, where \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature.
3. Since the internal energy depends only on temperature, the change in internal energy, \( \Delta E \), is calculated as \( \Delta E = E_2 - E_1 = \frac{3}{2}nR(T_2 - T_1) \).
4. Note that \( \Delta E \) is independent of the process or pathway taken between the two states, as it relies solely on the initial and final temperatures.
5. Conclude that for an ideal gas, the change in internal energy is a state function and does not depend on the path taken.

Key Concepts

Internal Energy

Internal energy is the total energy contained within a system, including kinetic and potential energies of the particles. For an ideal gas, the internal energy is primarily a function of temperature, meaning that it changes with temperature variations. In this case, as the gas temperature decreases from 300 K to 200 K, the internal energy will also decrease, reflecting the energy lost by the gas.

Path Independence of Internal Energy Change

The change in internal energy (ΔE) for a system is a state function, which means it depends only on the initial and final states of the system, not on the pathway taken to get there. This implies that regardless of how the gas transitions from state 1 to state 2, as long as the initial and final temperatures are the same, the change in internal energy will remain constant.

Ideal Gas Behavior

An ideal gas is a theoretical gas composed of many particles that are in constant random motion and do not interact with each other except during elastic collisions. The behavior of ideal gases is described by the ideal gas law, which relates pressure, volume, and temperature. Understanding this concept is crucial for analyzing changes in state, as it allows us to predict how internal energy and other properties will change with temperature and pressure.