In chemistry, the formula PV=nRT is the state equation for a hypothetical ideal gas. The ideal gas law describes the behavior of an ideal sample of gas, and how that behavior is related to the pressure (P), temperature (T), volume (V), and molarity (n) of the gas sample. In the equation PV=nRT, the term “R” stands for the universal gas constant.
The universal gas constant is a constant of proportionality that relates the energy of a sample of gas to the temperature and molarity of the gas. It is sometimes called the ideal gas constant, the molar gas constant. It is also sometimes called the Regnault constant, in honor of the French chemist Henri Regnault whose quantitative data was first used to precisely calculate the value of the constant. The currently accepted value for the universal gas constant R is:
R Constant = 8.3144598 J/mol·K
The unit for the gas constant is the joule per mol-kelvin. This can be read as “work per mol per degree” Essentially, the gas constant relates the molar amount of gas and temperature of the gas to the amount of kinetic energy in the gas. One can calculate the universal gas constant by dividing the product of the pressure and volume of a gas by the molarity and temperature of the gas:
R = PV/nT
Derivation Of The Ideal Gas Law
“Gases are distinguished from other forms of matter, not only by their power of indefinite expansion so as to fill any vessel, however large, and by the great effect heat has in dilating them, but by the uniformity and simplicity of the laws which regulate these changes.” — James Clerk Maxwell
The ideal gas law is one of the most fundamental equations in physical chemistry, and it has been independently derived through experimental analysis and theoretical extrapolation. Originally, the ideal gas law emerged as a combination of 4 other distinct mathematical expressions that relate various properties of a gas to one another. The four individual laws are: Charles’s law, Boyle’s law, Gay-Lussac’s law, and Avagadro’s law.
Charles’s law is an empirical law that states that the volume of a gas is directly proportional to the temperature of the gas. In other words, keeping all other factors equal, if one increases the temperature of a gas, they will observe a corresponding increase in the volume of the gas. Likewise, if one lowers the temperature of a gas, they will see a corresponding decrease in the volume. Mathematically, Charles’s law can be written as:
- V ∝ T
where “∝” means “directly proportional to”, or
- V/T = constant
Essentially, Charles’s law is a mathematically precise way of stating the oft-observed fact that gases tend to expand when heated.
Boyle’s law is a gas law that describes how the pressure of a sample of gas tends to increase as the volume of that sample decreases. Boyle’s law can be stated as “the pressure of a gas in a closed system at a constant amount and temperature is inversely proportional to the volume of the gas.” Mathematically, this can be written as:
- V ∝ 1/P
- PV = constant
Boyle’s law basically tells us that if we compress a gas, it has less space to be in and so it pushes harder on the walls on its container.
Gay-Lussac’s law is an empirical generalization that notes the relationship between the temperature of a sample of gas and its pressure. Gay-Lussac’s law states “at a constant volume and amount, the pressure of a gas is directly proportional to the temperature of the gas. This law can be written mathematically as:
- P ∝ T
- P/T = constant
Basically, Gay-Lussac’s law tells us that if we heat up a sample of gas, we will see a corresponding increase in its pressure. Temperature is just a measure of molecular motion, so heating up a gas makes the constituent particles move faster. The faster the constituent molecules move, the more force they will exert against the walls of the container — the gas will exert greater pressure. Gay-Lussac’s law offers an explanation for why heating a sealed container of gas can blow the container up; the pressure exerted by the gas becomes too great for the material to handle and it ruptures.
The last of the 4 pieces to the ideal gas equation is Avagadro’s law. Avagadro’s law states that the volume of a gas at constant pressure and temperature is directly proportional to the number of particles that make up the gas. Another way of stating the law is that if 2 samples of gas have the same volume at a constant temperature and pressure, then the 2 samples of gas have an identical number of particles. The equation for Avagadro’s law is:
- V ∝ n
where n is the number of individual particles. Avagadro’s law can also be written as:
- V/n = constant
Avagadro’s law is very intuitive. It is common sense that, all other things being equal, the more of gas there is, the more space it will take up. Alternatively, if two gases have the same volume, they must have the same amount of particles.
Deriving The Ideal Gas Law
Now that we have the 4 fundamental state equations for gas, we can combine them into one single expression to yield the ideal gas law. We can combine laws like this:
- V ∝ T (Charles’s law)
- V ∝ 1/P (Boyle’s law)
- P ∝ T (Gay-Lussac’s law)
- V ∝ n (Avagadro’s law)
Combining these expressions gives us:
- V ∝ nT/P
Since “∝” represent direct proportionality, we can replace the “∝” with a “=” by adding a constant of proportionality to the right-hand side. Experimentally, we have verified this constant to be equal to the value of R, so adding R to the equation yields::
- V = nRT/P
Rearranging this equation gives us:
- PV = nRT
The Significance Of The R Constant
“Poets say science takes away from the beauty of the stars – mere globs of gas atoms. I, too, can see the stars on a desert night, and feel them. But do I see less or more?” — Richard P. Feynman
So what exactly is the universal gas constant? The other parameters in the ideal gas equation all seem to correspond to some physically significant variable; pressure (P), volume (V), amount of a substance (n), and temperature (T). R however, does not seem to do this. As with many mathematical constants, the term R does not explicitly map onto some physical quantity, entity, or process. Instead, the parameter R represents a relationship that holds between some physical quantities, specifically the pressure and volume of a gas, and the temperature and amount of gas. Specifically, R is equal to the ratio PV/nT.
The exact numerical value of the gas constant actually varies with the chosen units. The numerical value of R as 8.3144598 is a result of the specific units we use. This value of R is a result of measuring the physical magnitudes of gases in the standard SI units. The standard SI units and their symbol for each parameter in the ideal gas equation are:
- Pressure (P) – Newtons (kg·m/s²)
- Volume (V) – Meter (m³)
- Temperature (T) – Kelvin (K)
- Amount of substance (n) – moles (mol)
If we changed our units, then the numerical value of the gas constant will also change. For example, say we decided to measure the volume of gas in liters (L) instead of meters, and the pressure of the gas in standard atmospheres (atm) instead of Newtons. With these units, the universal gas constant takes on a numerical value of R = 0.082057 L·atm/mol·K. Likewise, say we decide to measure pressure in terms of millimeters of mercury (mmHg). Then, the gas constant takes on a numerical value of R = 62.3636711 m³·mmHG/mol·K
It is important to realize that changing the units does not mean that the gas constant itself changes. The gas constant is just that, a constant, and so it does not change. Changing the units merely changes the numerical value used to express the constant. Theoretically, it would be possible to choose a unit system that changes the numerical value of the gas constant to 1. In such a unit system, the ideal gas equation could just be written as PV = nT. Keep in mind though that in this equation, the universal gas constant has not disappeared. The gas constant is still present, it just has a numerical value of R = 1. The constant itself is still required to give the appropriate dimensional analysis of the units used.
In essence, the parameter R represents a relationship that holds between the physical parameters of gas, and the units that we choose to measure those physical parameters. Therefore, the gas constant can be used to convert the physical measurements of gas into different unit systems.
Limitations Of Ideal Gas Law
There is a reason it is called the “ideal” gas law instead of the “actual” gas law. The validity of the ideal gas equation depends on a handful of idealized assumptions about the character and behavior of gases. First, the ideal gas law assumes that particles in a gas obey Newton’s laws of mechanics. This means that gas particles are assumed to obey the laws of force and gravity described by Isaac Newton and the effects of electrostatic intermolecular attractions are not considered.
“Today’s science fiction is tomorrow’s science fact.” — Isaac Asimov
Second, it is assumed that the molecules of the gas are negligibly small compared to the entire volume of the gas. This assumption allows scientists to simplify their calculations for the volume by leaving out the non-zero volume that molecules actually have.
Thirds, collisions between the molecules and the walls of the container are considered to be perfectly elastic—that is, no kinetic energy is lost from collisions. In actuality, a tiny amount of kinetic energy is absorbed by the walls of the container and is dissipated as heat. Normally, this tiny amount of energy is negligible and can be ignored.
Because of these assumptions, the “universal” gas law is technically not universal and is only accurate across a certain scope. Specifically, in a very cold sample of gas, intermolecular interactions overcome the kinetic energy of particles, which makes the behavior of the gas deviated from the ideal behavior. More complex state equations, such as van der Waals equations, are used to account for the effects on the behavior of particles due to intermolecular forces.