Question

The quantity $\dfrac{{h\nu }}{{{k_B}}}$ corresponds to:
a.) Wavelength
b.) Velocity
c.) Temperature
d.) Angular momentum

Answer 1

Hint: Find the dimensional formula of the terms in $\dfrac{{h\nu }}{{{k_B}}}$ , i.e. of h, v and Boltzmann constant, ${k_B}$ and then solve the question further by putting the dimension in $\dfrac{{h\nu }}{{{k_B}}}$ .

Complete step-by-step answer:
We have the quantity $\dfrac{{h\nu }}{{{k_B}}}$
Here, h is the Planck’s constant, v is the frequency and ${k_B}$ is the Boltzmann constant.
Now, to find the dimension of Planck’s constant, we know that $E = h\nu $ , here E is the energy and $\nu $ is the frequency.
Dimensional formula of energy is $[E] = [M{L^2}{T^{ - 2}}]$ and frequency is $[\nu ] = [{T^{ - 1}}]$
So, the dimension of h will be-
$[h] = \dfrac{{[E]}}{{[\nu ]}} = \dfrac{{[M{L^2}{T^{ - 2}}]}}{{[{T^{ - 1}}]}} = [M{L^2}{T^{ - 1}}]$
We know that the dimension of frequency is –
$[\nu ] = [{T^{ - 1}}]$
Now, the dimension of Boltzmann constant is to be find out-
Boltzmann constant is equal to Energy / Temperature, i.e., ${k_B} = \dfrac{E}{{Temp.}}$
Dimensional formula of energy is $[E] = [M{L^2}{T^{ - 2}}]$ and temperature is $[K]$
So, the dimensional formula will be $[{k_B}] = \dfrac{{[M{L^2}{T^{ - 2}}]}}{{[K]}} = [M{L^2}{T^{ - 2}}{K^{ - 1}}]$
Now, putting the dimensional formulas of Planck’s constant, frequency and Boltzmann constant in $\dfrac{{h\nu }}{{{k_B}}}$ , we get-
\[\dfrac{{[h][\nu ]}}{{[{k_B}]}} = \dfrac{{[M{L^2}{T^{ - 1}}][{T^{ - 1}}]}}{{[M{L^2}{T^{ - 2}}{K^{ - 1}}]}} = \dfrac{{[M{L^2}{T^{ - 2}}]}}{{[M{L^2}{T^{ - 2}}{K^{ - 1}}]}} = [K]\]
So, we obtained the dimension of the given quantity as [ K], which is temperature.
Hence, the quantity $\dfrac{{h\nu }}{{{k_B}}}$ corresponds to temperature.
Therefore, the correct option is C.

Note – Whenever such types of questions appear, then you should be aware of the term dimensional formulas, as these are very useful in knowing what a particular quantity corresponds to. As mentioned in the solution, first we try to find the dimensional formula of the given quantity. To find that, we have obtained the dimensional formula of all the terms appearing in the quantity, one by one and then substitute in the quantity, and then we obtained that the given quantity corresponds to temperature.