# The pVT properties of substances

## State of matter

There are generally three states of matter in nature, namely solid, liquid, and gas.

Regardless of the state of matter, there are many macroscopic properties, such as pressure (p), volume (V), temperature (T), internal energy (U), and so on. Since the liquid, solid, and other aggregated substances have little change in volume when the temperature and pressure change, but other changes are obvious, so only the pVT relationship of gas will be discussed below.

## Equation of state for an ideal gas

Under normal temperature and pressure, the higher the temperature of the gas and the lower the pressure, that is, the thinner the gas, the better the accuracy. Under this condition, the average distance between gas molecules is very far, the force between molecules can be ignored, and the volume of the molecule itself is also negligible compared with the distance between molecules.

Ideal gas: For the needs of theoretical research, the concept of an ideal gas model is established, that is, a gas that can strictly obey the basic laws of gas at any temperature and pressure is called an ideal gas.

### The role of the ideal gas concept

An ideal gas is a hypothetical gas in which the molecules themselves have no volume and no interaction force between molecules. Although ideal gases do not exist in nature, under high temperatures and low pressure, the molecular distance of real gas is very large, the force is small, and the volume of the molecule itself is negligible compared with the distance between gas molecules. In this way, the concept of an ideal gas can be used to derive relevant formulas to calculate the physical quantities of real gases.
Even at normal temperature and pressure, treating the real gas(such as supercritical co2) as an ideal gas and using the laws of the ideal gas to solve some practical problems will not cause too much error.

### Ideal gas equation of state

An ideal gas obeys the fundamental laws of gases at any temperature and pressure. Suppose 1 mol of ideal gas changes from state A (PA, TA, VMA) to state B (PB, TB, VMB). This change process can be envisaged to be carried out in two steps, as shown in Figure 1. First, expand from state A to state C at a constant temperature, then compress to state B at constant pressure, as shown in Figure 2.

## Real gases and the van der Waals equation

### The reason for the difference between ideal gas and real gas

• The ideal gas molecule itself has no volume, but the molecular volume of real gas exists, but the gas is very thin at high temperature and low pressure, and the gas molecular volume is negligible compared with the distance between molecules; but at high pressure and low temperature, the gas volume decreases, the molecular distance decreases and the gas molecular volume cannot be ignored.
• There is no force between ideal gas molecules, but there is a force between actual gas molecules, and intermolecular attraction is the main force. When the temperature is high, due to the intense molecular motion, the kinetic energy of the molecules is relatively large. In contrast, the force between molecules can be ignored; at the same time, when the pressure is low, the gas density is small, the distance between molecules is large, and the molecule’s gravitational force between them can also be ignored. However, the intermolecular forces cannot be ignored at low temperatures or high pressure.

### Van der Waals equation

The van der Waals equation is the most representative of the real gas equation of state. This equation contains two correction terms volume and intermolecular attraction.

Volume correction term: According to the ideal gas state equation, for a 1mol body, pVₘ=RT. In the formula, Vm refers to the free movement space of gas molecules in the current state. For an ideal gas, since molecules themselves have no volume, Vm is equal to the volume of the container. For a real gas, because the molecule itself has a volume, the free space for 1 mol of gas molecules to move freely is no longer Vm, but a correction term constant b related to the volume of the gas molecule itself must be subtracted from Vm, that is, Vm is replaced by Vₘ-b, the constant b is related to the type of gas.

### Correction term for intermolecular attraction

In the ideal gas state equation pVₘ=RT, the pressure refers to the pressure generated by the gas molecules colliding with the container wall at the temperature T. There is no gravitational force between the molecules. For real gases, due to the existence of intermolecular gravitation, there is a certain restriction on molecular motion, and the pressure on the container wall due to collisions is smaller than that of ideal gases.

If the pressure shown by the natural gas is p, the ideal gas pressure when there is no gravitational force between molecules is:
P+PB. PB is approximately proportional to the size of the intermolecular attraction, and also proportional to the number of molecules colliding on the wall of the unit container. The above two factors are inversely proportional to the molar volume, so PB should be inversely proportional to the square of the molar volume Vₘ, and the proportionality constant is represented by a.

The a and b in the above two formulas are physical constants related to the gas type, collectively referred to as the van der Waals constant of the gas. They are respectively related to the force between gas molecules and the size of the molecular volume. The units of a and b are Pa·m⁶/mol² and m³/mol, respectively.

The van der Waals equation is a semi-theoretical and semi-empirical equation of state. The constants a and b are data measured through real gas experiments. The pVT calculated by the van der Waals equation is much more accurate than that calculated by the ideal gas state equation. The pressure applied to this equation can reach the medium pressure range of several MPa; when the pressure is higher, there is a large deviation between the calculation result of the van der Waals equation and the experimental measurement value.