Internal pressure
Internal pressure is a measure of how the internal energy of a system changes when it expands or contracts at constant temperature. It has the same dimensions as pressure, the SI unit of which is the pascal. Internal pressure is usually given the symbol formula_1. It is defined as a partial derivative of internal energy with respect to volume at constant temperature: formula_2 Thermodynamic equation of state. Internal pressure can be expressed in terms of temperature, pressure and their mutual dependence: formula_3 This equation is one of the simplest thermodynamic equations. More precisely, it is a thermodynamic property  relation, since it holds true for any system and connects the equation of state to one or more thermodynamic energy properties. Here we refer to it as a "thermodynamic equation of state." Perfect gas. In a perfect gas, there are no potential energy interactions between the particles, so any change in the internal energy of the gas is directly proportional to the change in the kinetic energy of its constituent species and therefore also to the change in temperature: formula_4. The internal pressure is taken to be at constant temperature, therefore formula_5, which implies formula_6 and finally formula_7, i.e. the  internal energy of a perfect gas is independent of the volume it occupies. The above relation can be used as a definition of a perfect gas. The relation formula_7 can be proved without the need to invoke any molecular arguments. It follows directly from the thermodynamic equation of state if we use the ideal gas law formula_9. We have formula_10 Real gases. Real gases have non-zero internal pressures because their internal energy changes as the gases expand isothermally - it can increase on expansion (formula_11, signifying presence of dominant attractive forces between the particles of the gas) or decrease (formula_12,dominant  repulsion). In the limit of infinite volume these internal pressures reach the value of zero: formula_13, corresponding to the fact that all real gases can be approximated to be perfect in the limit of a suitably large volume. The above considerations are summarized on the graph on the right. If a real gas can be described by the van der Waals equation of state formula_14 it follows from the thermodynamic equation of state that formula_15 Since the parameter formula_16 is always positive, so is its internal pressure: internal energy of a van der Waals gas always increases when it expands  isothermally. The formula_16 parameter models the effect of attractive forces between molecules in the gas. However, real non-ideal gases may be expected to exhibit a sign change between positive and negative internal pressures under the right environmental conditions if repulsive interactions become important, depending on the system of interest. Loosely speaking, this would tend to happen under conditions such that the compression factor of the gas is greater than 1. In addition, through the use of the Euler chain relation it can be shown that formula_18 Defining formula_19 as the "Joule coefficient" and recognizing formula_20 as the heat capacity  at constant volume formula_21, we have formula_22 The coefficient formula_23 can be obtained by measuring the temperature change for a constant-formula_24 experiment, i.e., an adiabatic free expansion (see below). This coefficient is often small, and usually negative at modest pressures (as predicted by the van der Waals equation). The Joule experiment. James Joule tried to measure the internal pressure of air in his expansion experiment by adiabatically pumping high pressure air from one metal vessel into another evacuated one. The water bath in which the system was immersed did not change its temperature, signifying that no change in the  internal energy occurred. Thus, the internal pressure of the air was apparently equal to zero and the air acted as a perfect gas. The actual deviations from the perfect behaviour were not observed since they are very small and the specific heat capacity of water is relatively high. References. Peter Atkins and Julio de Paula, "Physical Chemistry 8th edition", pp. 60–61