Class 11th Physics Units and Measurement

The work done by a gas molecule in an isolated system is given by $W = α β^{2} e^{- \frac{x^{2}}{α k T}}$ , where x is the displacement, k is the Boltzmann constant and T is the temperature. α and β are constants. Then the dimensions of β will be:

Option 1 - <p><sup><span class="mathml" contenteditable="false"> <math> <mrow> <mrow> <mo>[</mo> <mrow> <mi>M</mi> <mi>L</mi> <msup> <mrow> <mi>T</mi> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mrow> </math> </span></sup></p>

Option 2 - <p><span class="mathml" contenteditable="false"> <math> <mrow> <mrow> <mo>[</mo> <mrow> <mi>M</mi> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi>T</mi> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mrow> </math> </span></p>

Option 3 - <p><span class="mathml" contenteditable="false"> <math> <mrow> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mi>M</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mi>L</mi> <msup> <mrow> <mi>T</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mrow> </math> </span></p>

Option 4 - <p><span class="mathml" contenteditable="false"> <math> <mrow> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mi>M</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msup> <mi>L</mi> <msup> <mrow> <mi>T</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mrow> </math> </span></p>

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Answered by

Vishal Baghel | Contributor-Level 10

4 months ago

Correct Option - 1

Detailed Solution:

Since, $\frac{x^{2}}{α k T}$ $\frac{^{}}{}$ should be dimensionless.

So, dimension of $α, [α] = \frac{L^{2}}{M L^{2} T^{- 2}} = M^{- 1} T^{2}$ $\frac{^{}}{^{}^{}}^{}^{}$

Dimension of $α β^{2}$ $^{}$ should be that of W.

So, $[α β^{2}] = M L^{2} T^{- 2}$ $^{}^{}^{}$

$\Rightarrow [β^{2}] = \frac{M L^{2} T^{- 2}}{M^{- 1} T^{2}} = M^{2} L^{2} T^{- 4} \Rightarrow [β] = M L T^{- 2}$

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