Dimensional Formula of Universal Gas Constant

Dimensions: Dimensions of the physical quantity are the powers to which the base quantities are raised to represent that quantity.

Dimensional Formula

The dimensional formula of any physical quantity is that expression which represents how and which of the base quantities are included in that particular quantity. 

It is written by enclosing the symbols for base quantities raised to the appropriate powers in square brackets i.e

Example: The Dimension formula of mass is: M

Now before we go into the explanation of the universal gas constant it is very important to know other terms which are equally relevant and make up the base for understanding the gas constant. 

Heat of Vaporization Lv: we all know that heat is required to convert a unit of liquid at a given specific temperature into its vapour form so heat of evaporation can be defined as the quantity of heat required to convert a unit of liquid at a specific temperature into its vapour at the same temperature. This value can be calculated and the value of this quantity is given at the normal boiling point of the liquid and can be measured in kJ/kg or Btu/lb.

Heat of Combustion Qf: The amount of heat generated by the complete combustion of one mole of a compound. It can also be understood by the given formula kj/kg or Btu/lb which is known as the heating value.  

Thermal conductivity which is denoted by k can be described as The time rate of heat transfer by conduction through a unit thickness of material per unit difference of temperature.

It can be measured as W/m K or Btu/ft h R.

Vapor Pressure, pv: it is the pressure exerted by a vapor when a state of equilibrium is reached between a liquid, or a solid, and its vapor and is expressed in Pa, psia, or mmHg.

Surface tension, σ is measured in units of N/m of lb/ft (1N/m=0.0686 lb /ft).

When we talk about any gassiest state it is important to understand it in simpler words so the gaseous state can be said to be the most simplified state of matter. Given the nature of gas which is highly compressible in nature, however, it has a uniform distribution of pressure on all sides. One basic advantage of gaseous state is the fact of it being shapeless and without any fixed volume so they can take the shape of any container they are placed into. Gas can also be easily mixed due to its negligible interaction of the intermolecular forces. Is the characteristic trait of gases can be read and governed by a few sets of laws, the laws which was derived on the basis of the experimental studies which were conducted on the behavioural pattern of gas under various different conditions, there were several studies and experiments conducted where one would read and observe the behavioural pattern of gas under conditions like pressure, temperature and volume. 

Gas constant can also be referred by its other scientific names such as molar gas constant, universal gas constant and the ideal gas constant, which is denoted by the symbol R. Gas constant is considered as an important rateable factor of numerous laws and principles of physics, although it is similar to Boltzman constant by the way it is represented in energy per degree temperature increment per mole, but it can also be used as a combination constant in various laws and as fundamental factors in equation. 

When we talk about Boltzman constant we can define it as the proportional factor that relates the average relative kinetic energy of particles and the temperature is given in a gas. But we should take notice of the fact that in the case of the Boltzman constant instead of a small value which is the product of volume and pressure, particle value is used in its place.

Now we will see different other laws where the gas constant and its relatable factors and characteristic plays an important role :

• Boyle’s Law – in Boyles law we see that while the pressure (P) of a gas increases it results in a decrease in its volume(V). When Sean mathematically the P can be said to be inversely proportional to V.

• Charles Law – we all know that the volume of gas expands when there is a rise in temperature so Charles law states that gas volume expands due to the rising temperature T despite its constant pressure. When we derive this formula mathematically we see that V is proportional to T.

• Avogadro’s Law – this law is the simplest and yet the most distinctive law when defining the property of gases, this law states that equal volume of all gas at the same temperature and pressure had the same number of molecules present within them.

• Gay Lussac’s law – the most important and most state outlaw in the field of both chemistry and physics talks about the pressure of a given mass of gas that varies directly with the absolute pressure of the gas in the volume is kept constant.

Let's Look Into Ideal Gas Law in a More Detailed Manner:

When we look into the ideal gas law it is important to understand that it is a combination of all the laws that we have mentioned above which consists of Boyle's law, Charles law and Avogadro's law. Every gas has its own net features and properties so it is very difficult to define and confine the gases and these three laws. Therefore when we define an ideal gas we see that the combination of the three laws is present as well as the gases bind with the structural integrity of all gases only then it would be considered as a specimen data for analysis. Just as the name suggests ideal gas is a theoretical concept where is the representation of a gas that is composed of point particles that do not show any changes during into molecule movements can be seen and this gas represents all the three primary laws stated by boil, Charles, gay Lussac and Avogadro.

Dimensional Equation 

The equation obtained by equating a physical quantity with its dimensional formula is called a dimensional equation. 

Application of Dimensional Analysis 

1. To convert a physical quantity from one system of the unit to the other:

It is based on a fact that the magnitude of a physical quantity remains the same no matter whatever system is used for measurement, i.e, magnitude = numeric value(n) multiplied by unit (u) = constant.

n₁u₁= n₁u₂

2. To check the dimensional correctness of a given physical relation: 

If in a given physical relation, the terms on both sides of the equation have the same dimensions, then the equation is dimensionally correct. This concept is best known as the Principle of Homogeneity of Dimensions. 

3. To derive the relationship between different physical quantities: 

Using the principle of homogeneity of dimension the new relation among physical quantities can be derived if the dependent quantities are known. 

Limitation of Dimensional Analysis

1. This method can be used only if dependency is of multiplication type. The formula containing exponential, trigonometric, and logarithmic functions can not be derived using this method. A formula containing more than one term that is added or subtracted like: s = ut+ ½ at2 cannot be derived either.

2. The relation derived from this method gives no information about the dimensionless constants. 

Universal Gas Equation

The terms pressure, volume, and temperature of a gas are all interrelated and this interrelation was first explored by Robert Boyle. Boyle’s law states that when the sample gas temperature is kept constant, PV = Constant.

French scientist Jacques Charles came up with his law later named Charles law. He discovered when the pressure of a gas is kept constant, the volume of the gas can be related to the temperature by the following equation.

V/T = Constant.

These laws were later combined to yield one universal gas law known as the ideal gas law.

PV/T=constant

And the constant proportionality factor in this equation is the Universal Gas Constant, R ie, constant = nR, where, n gives the number of moles of the gas in the sample.

Thus, According to Boyle's law volume of a gas is inversely proportional to its pressure at constant temperature and number of moles. 

According to Charles law volume is directly proportional to temperature at constant pressure and number of moles. 

Avogadro volume is directly proportional to the number of moles at a constant temperature and number of moles. 

So by combining all these laws we get a universal gas equation, i.e, PV=nRT

Derivation of Gas Equation:

V=nRT/P 

By the above equation, PV=nRT,

R=PV/nT=Constant

Here R is the universal gas constant.

Dimensional Formula of the Universal Gas Constant

R= Pressure.volume/Temperature moles

So its unit= litre.atm/ k mol

The dimension of the universal gas constant will be:

R= [M¹ L² T^-2 K^-1]

Value of R= 0.0821 litre atm/ k mol.