Internal pressure

Internal pressure is a specific way to measure energy changes in a system. It describes how internal energy shifts when a system expands or contracts. This process must happen at a constant temperature to be measured this way. Scientists use the symbol $\pi_T$ to represent this property. The standard unit for this measurement is the pascal. This value is important because it connects energy to the physical volume of a system. It is known as a thermodynamic equation of state. This means it links the state of a system to its energy properties.

To understand the mechanism, we look at how energy and volume interact. The internal pressure is defined as a partial derivative of internal energy with respect to volume. This calculation occurs while keeping the temperature constant. We can also express this value using temperature and pressure. The math involves the fundamental thermodynamic equation. By dividing this equation by volume at a constant temperature, we can find the internal pressure. We also use Maxwell relations to help solve these complex equations. These steps allow scientists to predict how energy will move within a system.

Different types of gases behave in very different ways. We can categorize them into perfect gases and real gases. In a perfect gas, the particles have no potential energy interactions. This means there are no forces pulling or pushing the particles together. Because of this, any change in internal energy is only linked to temperature. If the temperature stays constant, the internal energy does not change when volume changes. Therefore, the internal pressure of a perfect gas is zero. This is a very useful way to define what a perfect gas is.

Real gases are more complex because they have non-zero internal pressures. Their internal energy changes as they expand at a constant temperature. This happens because of the forces between the gas particles. If attractive forces are dominant, the internal energy increases during expansion. If repulsive forces are dominant, the internal energy decreases instead. As the volume becomes infinitely large, these internal pressures eventually reach zero. At this point, the real gas begins to act like a perfect gas.

We can use the van der Waals equation to model these real gases. This equation includes a specific parameter to account for molecular attraction. This parameter is always a positive value. Because of this, the internal pressure of a van der Waals gas is always positive. This means the internal energy of such a gas always increases during isothermal expansion. However, other real gases might show different results. Under specific temperatures and pressures, repulsive interactions might become more important. This could cause the internal pressure to change signs.

History shows us how scientists discovered these subtle energy shifts. James Joule conducted an expansion experiment to measure the internal pressure of air. He used an adiabatic process to pump air between two metal vessels. One vessel was evacuated to allow for expansion. The system sat in a water bath to keep the temperature steady. Joule observed no change in temperature during the experiment. He concluded that the internal pressure of air was zero. He believed the air acted as a perfect gas.

Later researchers found that Joule's results were not entirely complete. In 1925, Frederick Keyes and Francis Sears used better tools to study carbon dioxide. They performed measurements at 30 °C and pressures between 13.3 and 16.5 atm. They looked at the Joule coefficient, which is related to internal pressure. This coefficient can be measured by watching temperature changes during adiabatic free expansion. They found that the temperature dropped when the pressure was lowered. This showed that the internal pressure for carbon dioxide was negative. This discovery matched the predictions made by the van der Waals equation.