state the dimension of universal gravitational constant
Answers
Explanation:
r, G = [M1 L1 T-2] × [L]2 × [M]-2 = [M-1 L3 T-2]. Therefore, the gravitational constant is dimensionally represented as M-1 L3 T-2.
The dimensional formula of Universal Gas Constant is given by,
M1 L2 T-2 K-1
Where,
M = Mass
L = Length
T = Time
Derivation
Since, P × V = nRT
Therefore, R = P × V × [nT]-1 . . . . . (1)
Where, P = pressure, V = volume, N = number of moles, T = temperature, R = Universal Gas Constant
The dimensional formula of volume = [M0 L3 T0] . . . . (2)
And, the dimensional formula of temperature = [M0 L0 T0 K1] . . . . (3)
Since, Pressure = Force × [Area]-1
= Mass × acceleration × [Area]-1 = [M] × [L1 T-2] × [L2]-1
∴ the dimensional formula of pressure = [M1 L-1 T-2] . . . . (4)
On substituting equation (2), (3) and (4) in equation (1) we get,
Universal Gas Constant = Presure × Volume × [n × Temperature]-1
Or, G = [M1 L-1 T-2] × [M0 L3 T0] × [M0 L0 T0 K1]-1 = [M1 L2 T-2 K-1].
Therefore, the Universal Gas Constant is dimensionally represented as [M1 L2 T-2 K-1]
hope it helps