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. a and b are constants. Then the dimensions of b 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 a month ago

Asked by Shiksha User

1 Answer

Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 1

Detailed Solution:

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

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

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

So, $[α β^{2}] = M L^{2} T^{- 2}$

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

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> 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> Dimension of  <math> <mrow> <mi>α</mi> <msup> <mrow> <mi>β</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </math> <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1818922362"> </o:OLEObject></xml><! [endif]->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> <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. a and b are constants. Then the dimensions of b 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. a and b are constants. Then the dimensions of b 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. a and b are constants. Then the dimensions of b will be: