Understanding the Change in Internal Energy of a Gas Expanding at Constant Temperature and Pressure

Internal energy (U) of a gas is an important thermodynamic property that represents the total energy contained within the gas. In this article, we will explore what happens to the internal energy of a gas when it expands at a constant temperature and pressure.

Expansion at Constant Temperature and Pressure: A Fundamental Insight

When a gas expands at constant temperature and pressure, it is crucial to understand how this affects its internal energy. According to the First Law of Thermodynamics, the change in internal energy ((Delta U)) is given by:

[Delta U Q - W]

where (Q) is the heat added to the system and (W) is the work done by the system. In this scenario, the temperature remains constant ((T text{constant})), and so does the pressure ((P text{constant})).

Work Done by the Gas During Expansion

During expansion, the gas does work as it pushes against an external pressure. The work done by the gas is given by:

[W P Delta V]

where (Delta V) is the change in volume. Since the pressure is constant, the work done can be expressed as proportional to the volume change.

Heat Added to the System

Given that the temperature remains constant, we can infer that no heat ((Q)) is added to the system as it expands. This is in accordance with the fact that an isothermal process at constant pressure involves no net heat transfer. Therefore, the heat added to the system is zero ((Q 0)).

Proportionality between Internal Energy and Volume

Given that (Q 0) and (W) is non-zero, the change in internal energy can be represented as:

[Delta U Q - W 0 - W -W]

Since (W P Delta V), we can conclude that the change in internal energy is directly proportional to the negative of the volume change:

[Delta U propto -Delta V]

However, the sign of (Delta U) indicates that the internal energy increases as the volume increases.

Implications for Mole Number

The increase in the number of moles (n) of gas as the system expands can be linked to the internal energy. The internal energy of an ideal gas is given by:

[U frac{3}{2} nRT]

where (R) is the universal gas constant. Since (T) and (P) are constant, the internal energy is directly proportional to the number of moles (n). Therefore, the increase in volume ((Delta V)) is associated with an increase in the number of moles, leading to an increase in internal energy.

Conclusion: Internal Energy and Expansion at Constant Temperature and Pressure

In conclusion, when a gas expands at a constant temperature and pressure, the internal energy does not remain constant. Instead, it increases in proportion to the volume increase. This increase in internal energy is directly related to the increase in the number of moles of gas added to the system.

Understanding these principles is crucial for anyone working with thermodynamics, whether in academic or industrial settings. By maintaining a balance between temperature, pressure, and volume, we can effectively manage and harness the internal energy of gases in various applications.