Question

The average kinetic energy $E$ of a gas molecule is given by the expression $E = \dfrac{3}{2}kT$ where $T$ is the absolute (Kelvin temperature).
What are the SI base units of $k$?
(A) $k{g^{ - 1}}{m^{ - 1}}{s^2}K$
(B) $k{g^{ - 1}}{m^{ - 2}}{s^2}K$
(C) $kgm{s^{ - 2}}{K^{ - 1}}$
(D) $kg{m^2}{s^{ - 2}}{K^{ - 1}}$

Answer 1

The problem can be solved by separating k in the given equation and writing it in the form of other variables. Then writing each quantity in terms of the SI base units of the fundamental quantities will give out the required result.

Complete step by step answer:
The given expression is
$\Rightarrow E = \dfrac{3}{2}kT$
Or $k = \dfrac{{2E}}{{3T}}$ (1)
Now we must write each quantity in its SI base units
Kinetic Energy, $E = kg{m^2}{s^{ - 2}}$
Temperature, $T = K$
$\Rightarrow \dfrac{2}{3}$ is a constant, hence it has no units
Substituting these in (1), we get
$\Rightarrow k = \dfrac{{kg{m^2}{s^{ - 2}}}}{K}$
On solving we get:
$\Rightarrow k = kg{m^2}{s^{ - 2}}{K^{ - 1}}$
Hence the correct answer is option (D); $kg{m^2}{s^{ - 2}}{K^{ - 1}}$.

Additional Information
All the quantities of the physical world have their units and dimensions in terms of the combination of some fundamental or base quantities. These are the seven fundamental quantities in terms of which the dimensions of other physical quantities are expressed. These, along with their SI base units and dimensions are:
-Mass – kg (kilogram) – [M]
-Length – m (meter) – [L]
-Time – s (second) – [T]
-Electric Current – A (Ampere) – [A]
-Temperature – K (Kelvin) – [K]
-Luminous intensity – cd (candela) – [cd]
-Amount of substance – mol (mole) – [mol]
(Square brackets [ ] round a quantity represents the dimensions of that quantity).

Note:
For solving such types of questions, it is required to obtain each physical quantity given in the problem in terms of the fundamental or base quantities. So we must know the physical formula which can relate the quantity with the fundamental quantities. It must be noted that there can be multiple formulas and any one of them can be used for this purpose.