n\(\propto\) I^aT^bm^c, [I] = [M^oL¹T^o]
[T] = [M¹L¹T^-2] (force)
[M] = [M¹L^-1T^o]
[M^o L^oT^-1] = [M^oL¹T^o]^a [M¹L¹T^-2]^b [M^oL^-1T^o]^C
b + c = 0
a + b - c = 0
-2b = -1 \(\Rightarrow\) b = \(1\over2\)
c =\(-{1\over2}a=1\)
n\(\propto\) \({1\over l}{\sqrt{T\over m}}\)

A^o= 10^-10 m
Atomic volume of 1 mole of hydrogen
= Avagadro's number\(\times\)volume of hydrogen molecule
= 6.023\(\times\)10²³ \(\times\)\(4\over 3\) \(\times\) \(\pi\) x (10^-10 m)³
= 25.2\(\times\)10^-7 m³
Molar volume = 22.4 L = 22.4\(\times\)10^-3 m³
\(Molar \ volume \over Atomic\ volume\)= 0.89x10⁴\(\approx \)104
This ratio is large because actual size of gas molecule is negligible in comparison to the inter molecular separation.

Let  T = Kr^aM^bG^c  ..... (1)
Where  K = a dimensionless constant
dimensions of the various quantities are [T] = T, [r] = L, [M] = M
[G] \(={Fr^2\over m_1m_2}={MLT^{-2}.L\over MM}=M^{-1}L^3T^{-2}\)
Substituting these dimensions in equation (1) we get,
[T] = [L]^a [M]^b [M^-1 L³ T^-2]^c
M^OL^oT^I=M^b-c=M^b-cL^a+3cT^-2c
Equating the dimensions of M, L and T, we get b - c = 0, a + 3c = 0, - 2c = 1
on solving a=\(3\over2\) b=\(-{1\over2}\) c=\(-{1\over2}\) T=Kr^3/2M^-1/2 G^-l/2 or T²=\({K^2R^3\over MG} \Rightarrow \therefore T^2 \propto r^3\)

(i) Given \(\lambda ={h\over mv}\)
As wavelength is a distance, 
LHS = \(\therefore [\lambda]=L\)
Also, RHS =\([{h\over mv}] ={Planck's \ constant \over mass \times velocity}={ML^2T^{-1}\over MLT^{-1}}=L\)
\(\therefore\) Dimensions of LHS = Dimensions of RHS.
Hence the given equation is dimensionally consistent.
(ii) Given  v = \(\sqrt{2GM\over R}\)
LHS = [V] = LT^-1
\(RHS=[{2GM\over R}]^{1\over2}=[{M^{1\over2}L^3T^{-2}.M \over L}]^{1\over 2}=[L^2T^{-2}]{1\over2}=LT^{-1}\)
\(\therefore\) Dimensions of LHS = Dimensions of RHS.
Hence the given equation is dimensionally correct.

As F \(=G{m_1m_2\over r^2}\)
\(G={F.r^2\over m_1m_2}\)
\([G]={{MLT}^{-2}.L^2\over MM}={M}^{-1}L^3{T}^{-2}\)
\(\therefore\) a = -1, b ~ 3, c = -2

\(\therefore\) \(n_2=n_1{\left[ {M_1 \over M_2} \right]}^{a}{\left[ {L_1 \over L_2} \right]}^{b}{\left[ {T_1 \over T_2} \right]}^{c}\)
\(=6.67\times{10}^{-8}{\left[ {{1\over 1000}} \right]}^{-1}{\left[ {{1\over 100}} \right]}^{3}\left[ {1\over 1} \right]^{-2}=6.67\times{10}^{}-11\)
Hence in SI units,   G = 6.67\(\times\)10^-11Nm² kg^-2