The work done by a gas molecule in an isolated system is given by <math> <mrow> <mi>W</mi> <mo>=</mo> <mi>α</mi> <msup> <mrow> <mi>β</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mfrac> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>α</mi> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </msup> </mrow> </math> , 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:

Since, x 2 α k T should be dimensionless.So, dimension of α , [ α ] = L 2 M L 2 T − 2 = M − 1 T 2 Dimension of α β 2 should be that of W.So, [ α β 2 ] = M L 2 T − 2 ⇒ [ β 2 ] = M L 2 T − 2 M − 1 T 2 = M 2 L 2 T − 4 ⇒ [ β ] = M L T − 2

The work done by a gas molecule in an isolated system is given by <math> <mrow> <mi>W</mi> <mo>=</mo> <mi>α</mi> <msup> <mrow> <mi>β</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mfrac> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>α</mi> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </msup> </mrow> </math> , 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:

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Physics Class 11th 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 -

^{$[M L T^{- 2}]$}

Option 2 -

$[M L^{2} T^{- 2}]$

Option 3 -

$[M^{2} L T^{2}]$

Option 4 -

$[M^{0} L T^{0}]$

2 Views | Posted 2 weeks ago

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1 Answer

Answered by

Vishal Baghel | Contributor-Level 10

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

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

Since, <math> <mrow> <mfrac> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>α</mi> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </math> <math><mrow><mfrac><mrow><msup><mrow></mrow><mrow></mrow></msup></mrow><mrow></mrow></mfrac></mrow></math> should be dimensionless.So, dimension of <math> <mrow> <mi>α</mi> <mo>, </mo> <mrow> <mo> [</mo> <mrow> <mi>α</mi> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <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> </mfrac> <mo>=</mo> <msup> <mrow> <mi>M</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mi>T</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </math> <math><mrow><mrow><mrow></mrow></mrow><mfrac><mrow><msup><mrow></mrow><mrow></mrow></msup></mrow><mrow><msup><mrow></mrow><mrow></mrow></msup><msup><mrow></mrow><mrow></mrow></msup></mrow></mfrac><msup><mrow></mrow><mrow></mrow></msup><msup><mrow></mrow><mrow></mrow></msup></mrow></math>Dimension of <math> <mrow> <mi>α</mi> <msup> <mrow> <mi>β</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </math> <math><mrow><msup><mrow></mrow><mrow></mrow></msup></mrow></math> should be that of W.So, <math> <mrow> <mrow> <mo> [</mo> <mrow> <mi>α</mi> <msup> <mrow> <mi>β</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <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> </math> <math><mrow><mrow><mrow><msup><mrow></mrow><mrow></mrow></msup></mrow></mrow><msup><mrow></mrow><mrow></mrow></msup><msup><mrow></mrow><mrow></mrow></msup></mrow></math> <math> <mrow> <mo>⇒</mo> <mrow> <mo> [</mo> <mrow> <msup> <mrow> <mi>β</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mfrac> <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> <mrow> <msup> <mrow> <mi>M</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mi>T</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <mo>=</mo> <msup> <mrow> <mi>M</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mi>T</mi> </mrow> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>⇒</mo> <mrow> <mo> [</mo> <mrow> <mi>β</mi> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mi>M</mi> <mi>L</mi> <msup> <mrow> <mi>T</mi> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </math>

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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:

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The work done by a gas molecule in an isolated system is given by W = α β 2 e − 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:

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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: