Dimensional Formula of Gas Constant

A gas constant is defined as the generic constant present in the equation of state of gases; it is determined by dividing the product of the volume and pressure of one mole by the absolute temperature in the case of an ideal gas. R is the gas constant, which is a physical constant expressed in units of energy per temperature increase per mole. Phrases such as universal gas constant, ideal gas constant, and the molar gas constant can be used interchangeably. The gas constant and Boltzmann constant are similar, but the former is determined by the product of volume and pressure while the latter is described as energy per increase in temperature per particle.

The ideal gas law can be written as

Pressure × Volume = Total number of moles × Temperature × Gas constant

Or PV = nRT

So, Gas constant = Pressure x Volume / Total number of moles x Temperature

Pressure = Force/Area = mass x acceleration / Area 

The dimensional formula of pressure P = [M1].[L1T-2][L2]

Simplifying the expression

Pressure P = [M1L1-2T-2] = [M1L-1T-2]

The unit of volume V is m3, so its dimensional formula is [L3]

The dimensional formula of temperature T is [K1]

Substitute the three dimensional formulae in the formula of gas constant

Gas constant R = [M1L-1T-2] x [L3][K1]

Gas constant R = [M1L-1T-2][L3][K-1]

Combining all the fundamental dimensional formulae

= [M1L-1+3T-2K-1]

= [M1L2T-2K-1]

So, the dimensional formula of gas constant is [M1L2T-2K-1].

Dimensional Formula

The expression indicating the powers to which fundamental units must be raised to obtain one unit of a derived quantity is the quantity’s dimensional formula.

Suppose Q is the unit of a derived quantity represented by the expression Q = MaLbTc. In that case, MaLbTc is the dimensional formula, and the exponents a, b, and c are referred to as the dimensions.

What are Dimensional Constants?

Dimensional constants are physical quantities that have dimensions and a fixed value. Examples include gravitational constant (G), Planck’s constant (h), the universal gas constant (R), and the speed of light in vacuum (c).

Law of Dimensional Homogeneity

● In any correct equation expressing the relationship between physical quantities, all terms must have the exact dimensions on both sides. Words denoted by a plus sign or a minus sign must have the exact dimensions.

● When a physical quantity Q has dimensions a, b, and c in length (L), mass (M), and time (T), respectively, and n1 is its numerical value in a system with fundamental units L1, M1, and T1, and n2 is its numerical value in another system with fundamental units L2, M2, and T2, , n1 [L1,a M1b T1 c ]= n2 [L2a M2b T2 c]

 Dimensional Analysis’s Limitations

● This method cannot be used to determine dimensionless quantities. This method cannot be used to determine the constant of proportionality. They can be discovered either experimentally or theoretically.

● This procedure does not work with trigonometric, logarithmic, or exponential functions.

● This method will be difficult to apply to physical quantities dependent on more than three physical quantities.

● In some instances, the constant of proportionality has dimensions as well. In such cases, we are unable to use this system.

● We cannot use this method to derive an expression of one side of the equation containing the addition or subtraction of physical quantities.

Coclusion

In the equation of state of gases, a generic constant is equal to the product of one mole’s pressure and volume divided by absolute temperature in the case of an ideal gas. The gas constant, R, is a physical constant that is quantified in units of energy per temperature increase per mole. The dimensional formula of gas constant is assessed by obtaining the dimensional equations of pressure, temperature, and volume as [M1L-1T-2], [K1], and [L3], the dimensional formula of gas constant was obtained to be [M1L2T-2K-1].