`The world is a dangerous place, not because of those who `

`do evil, but because of those who look on and do nothing. `

`–Albert Einstein `

Today we will discuss a few questions on units and dimensions:

(1) Van der Waals equation of state for real gases is [

*P*+ (*a/V*^{2})](*V*–*b*) =*RT*where*P*is the pressure,*V*is the volume of one mole of gas,*R*is the universal gas constant,*T*is the temperature and*a*and*b*are constants. The dimensional formula of*a*is(a) [M

^{0}L^{6}T^{0}](b) [ML

^{–1}T^{–2}](c) [ML

^{5}T^{–2}](d) [ML

^{–2}T^{–2}](e) [ML

^{3}T^{–2}]This is a question popular among question setters and has appeared many times in various entrance test papers. Since

*a/V*^{2}appears as added to*P*, the dimensions of*a/V*^{2}must be the same as those of*P*. Therefore, ‘*a*’ has the dimensions of*PV*^{2}. Note that pressure

*P*is force per unit ares and hence the dimensional formula for*P*is [ML^{–1}T^{–2}]. The dimensional formula for*V*^{2 }is [L^{6}]. Therefore, the dimensional formula for ‘*a*’ is [ML^{5}T^{–2}].(2) The dimensional formula for Stefan’s constant is

(a) [MT

^{–3}K^{–4}](b) [ML

^{2}T^{–2}K^{–4}](c) [ML

^{2}T^{–2}](d) [MT

^{–2}L^{0}](e) [MT

^{4}L^{0}]This question appeared in Kerala Engineering Entrance (KEAM) 2009 question paper. You will find similar questions in other entrance test papers also.

We have

*E =**σT*^{4}where*E*is the energy radiated*per second from unit area*,*σ*is Stefan’s constant and*T*is the temperature of the black body.Therefore,

*σ = E/T*^{4}Note that energy has dimensional formula [ML

^{2}T^{–2}]. Sine*E*is the*energy*radiated*per second from unit area*, the dimensional formula for*E*is [MT^{–3}].Therefore, the dimensional formula for Stefan’s constant

*σ*is [MT^{–3}K^{–4}].(3) Pick out the correct dimensions in

*length*of the quantity*μ*_{0}*ε*_{0 }from the following:(a) 2

(b) 1

(c) – 1

(d) – 2

(e) zero

You know that the speed ‘

*c*’*of electromagnetic waves in free space is given by**c*= 1/√(

*μ*

_{0}

*ε*

_{0})

Therefore

*μ*_{0}*ε*_{0}= 1/*c*^{2}and the dimensional formula for*μ*_{0}*ε*_{0}is [L^{–2}T^{2}]. The quantity*μ*_{0}*ε*_{0}therefore has – 2 dimensions in length [Option (d)].(4) Which of the following is dimensionless?

(a) Stress

(b) Gas constant

(c) Frequency of sound

(d) Efficiency of heat engine

(e) Thermal conductivity

The correct option is (d) since the efficiency is the ratio of the output power to the input power.