Dimensions of Gas constant

What is the Universal Gas Constant?

Dimensional Formula of Gas Constant  

The dimensional formula of Universal Gas Constant is represented as:

\[M^{1}L^{2}T^{-2}K^{-1}\]

Where,

  • M denotes the mass

  • L denotes the length

  • T denotes the time


Derivation of Gas Constant

Since, \[Pressure \times Volume = Total moles \times Temperature \times Gas Constant\]  

Therefore,  we calculate the dimension of the universal gas constant with the use of the following:

\[Gas Constant = Pressure \times Volume \times Total Moles \times Temperature\]

\[Total Moles \times Temperature^{-1}\] . . . .  (i)

The dimensional formula of temperature and volume = \[M^{0}L^{0}T^{0}K^{1} and M^{0}L^{3}T^{0} \]. . .(ii)

Since \[ Pressure = Force \times Area^{-1}\]

\[P = M \times a Area^{-1} = M \times LT^{-2} \times L^{-2}\]

Therefore, the dimensions of pressure = \[M^{1}L^{-1}T^{-2}\]

\[M^{1}L^{-1}T^{-2}\] . . . (iii)

On substituting equation (ii) and (iii) in equation (i) we get,

\[ Gas Constant = Pressure \times Volume \times Total Moles \times Temperature^{-1} \]

Or, \[ G = M^{1}L^{-1}T^{-2} × L^{3} \times K^{1} K^{1}-1 = M^{1}L^{2}T^{-2}K^{-1}\].

Therefore, the universal gas constant dimensional formula is represented as \[M^{1}L^{2}T^{-2}K^{-1}\]

Perfect Gas Equation of General Gas Equation and Specific and Universal Gas Constant

We know that,

According to Gay Lussacc’s law, P is directly proportional to T………(i)

According to Charle’s law, V is directly proportional to T…………..(ii)

Combining both the equations, PV is directly proportional to T or PV= Constant * T

If the mass of the gas is measured in kg or gm, the constant used is ‘k’, then,

PV= KT (Ideal Gas Equation)

Here, k is a gas constant depending upon the mass of a gas. 

For 1 kg of gas, K is called a specific gas constant if the mass of gas is 1 kg-mole or 1 gm-mole, then constant k will be the same as the value for all gases. 

Therefore, K is replaced by R, known as the universal gas constant.

Therefore, PV = RT--------(iv)

It is the general gas equation.

\[R = \frac{Mass * Length^{2}}{Amount * Temperature * (time)^{2}}\]

\[R = 8.314462618...kg.m^{2}.s^{-2}.k^{-1}.mol^{-1}\]


What is Universal Gas Constant?

Universal Gas Constant is also known as the molar gas constant, ideal gas constant, or universal gas constant. It is denoted by the symbol R. It is equivalent to the Boltzmann constant and can be expressed in units of energy per temperature per mole.


PV= nRT or PV= mRT


N is the number of moles and M is the mass


Universal Gas constant is named so because it is the same for all gases. It can also be derived from another theory called microscopic kinetic theory, as it was achieved (apparently independently) by August Krönig in the year 1856 and Rudolf Clausius in the year 1857.


Determination of Universal Gas Constant

Firstly, the experimental value of Boyle's law constant is determined with the help of the product of pressure P and volume V for a constant temperature T and a fixed mass m of air. After knowing these values, the universal gas law constant R is calculated from the equation PV = nRT.


What is Specific Gas Constant?

The specific gas constant of a gas or for a mixture of gases \[(R_{Specific})\] is represented by the molar gas constant divided by the molar mass (M) of the gas or mixture.


\[(R_{Specific = \frac{R}{M}})\]


Also known as the molar gas constant and the ideal gas constant, the universal gas constant is denoted by the symbol R. It is expressed in units of energy per temperature per mole and is equivalent to the Boltzmann Constant. It is represented as follows - 


PV = nRT where n is the number of moles or PV = mRT where m is the mass. 


It is called the “Universal” Gas Constant because it is constant across all gases. The microscopic kinetic theory also helps us derive the universal gas constant. This was achieved by August Kronig in 1856 and Rudolf Clausius in 1857. 


What is the Dimensional Formula of Gas constant?

The dimensional formula of Universal Gas Constant is given as - 

\[M^{1}L^{2}T^{-2}K^{-1}\]

Where, M = Mass, L = Length and T = time. 


How do we arrive at the Derivation of the Gas Constant?

We know that \[Pressure \times Volume = Total moles \times Temperature \times Gas Constant\]


Equation 1-

Hence, \[Universal Gas constant = Pressure \times Volume \times Total moles \times Temperature^{-1}\]


Equation 2- The dimensional formula for temperature is - \[M^{0}L^{0}T^{0}K^{1}\]

The dimensional formula for volume is - \[M^{0}L^{3}T^{0}\]

\[Pressure = Force \times Area^{-1}\]

Hence, \[ P = M \times a \times Area^{-1} = M \times LT^{-2} \times L^{-2}\]

Equation 3- Thus, we arrive at the dimensions of pressure which is represented as - \[M^{1}L^{-1}T^{-2}\]

When we substitute equations 1, 2, and 3, we arrive at - 

\[ Universal Gas Constant = Pressure \times volume \times total moles \times Temperature^{-1}\]

Or we say, \[G = M^{1}L^{1-}T^{-2} \times L^{3} \times K^{1}-1 = M^{1}L^{2}T^{-2}K^{-1}\]

Finally, the dimensional formula for universal gas constant is represented as - \[M^{1}L^{2}T^{-2}K^{-1}\]


What are Equations related to Gas Constant?

As per Gay Lussac’s Law, P and T are directly proportional to each other.. (i)

As per Charle’s Law, V and T are directly proportional to each other (ii)

When we combine both these equations, we realize that PV is directly proportional to T or rather PV = Constant x T

When we measure gas in kg or gm and the constant is taken as ‘k’ then, 

PV = KT which becomes an ideal gas equation. 

k here is defined as a gas constant that is dependent on the mass of a gas. 

Suppose we take 1 kg of gas. Then if the mass of the gas is 1 kg-mole or 1 gm-mole, we will say that K is a specific gas constant. We will thus say that the constant k is the same as a value for all the gases. 

Here we replace R for K as the universal gas constant. 

Thus, PV = RT (iv)

R= Mass∗Length (square) / Amount∗temperature∗(time) (square)

R = 8.314462618... \[kg.m^{2} -s^{-2}.K^{-1}.mol^{-1}\]


How do we Memorize these Formulae and Equations?

When we pick up physics as a subject, we come face to face with many equations and formulae which we are then required to apply to solve various physical problems which also form an important part of the final question paper! 


It is important to memorize these well, but more importantly, try to understand them better. After all, how many formulas can be rote learned? Isn’t it better to understand how they are derived? It also makes the learning process much more enjoyable! 


Vedantu has various resources to help students understand and memorize formulae and equations like an ace. Our revision notes are tailor-made to help enhance your memory. The key concepts are all in there as you sit to study! Students can also utilize important concepts and articles like these to strengthen their grasp of a topic. 


For this topic and many topics like these, you can actually use separate notebooks to note down key formulae and derivations with important points in the margins. This helps you retain the information and makes you an active learner. Also, it will aid you much beyond the examinations as well!

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