Dimensional Formula of Universal Gas Constant

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The dimension of the universal gas constant will be,

R= [M1 L2 T–2 K–1]

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].

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 This Method

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 at2cannot 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 as 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

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.

FAQ (Frequently Asked Questions)

1. Define the Dimension Formula.

Answer: Dimensions of the physical quantity are the powers to which the base quantities are raised to represent that physical quantity. The dimensional formula of any physical quantity is that expression which represents how and which of the base quantities are included in that physical quantity. 

It is written by enclosing the symbols for base quantities with appropriate power in square brackets i.e [ ].

2. Explain a Few Limitations of Dimension Formulas.

Answer: Some of the limitations of the dimension formula are given below.

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 which is added or subtracted likes s = ut+ ½ at2, also cannot be derived.


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

3. Write a Few Sets that have the Same Dimension Formula.

Answer: Some of the sets having the same dimension formula are discussed below.

1. Strain, refractive index, relative density, angle, solid angle, phase, distance gradient, relative permeability.

2. Mass and inertia.

3. Momentum and impulse.

4. Thrust, force, weight, tension, energy gradient.

5. Angular momentum and Planck’s constant.

6. Surface tension, surface area, force gradient, spring constant.

7. Latent heat and gravitational potential.

8. Thermal capacity, Boltzman constant, entropy.