The temperature of an ideal gas in 3-dimensions is 300K. The corresponding de-Broglie wavelength of the electron approximately at 300 K, is:

[m_e = mass of electron = 9 × 10^-³¹kg

h = Planck constant = 6.6 × 10^-³⁴ Js

k_B = Boltzmann constant = 1.38 × 10^-²³ JK^-¹]

Option 1 -

3.25 nm

Option 2 -

6.26 nm

Option 3 -

2.26 nm

Option 4 -

8.46 nm

6 Views | Posted 2 months ago

Asked by Shiksha User

1 Answer

Answered by

alok kumar singh | Contributor-Level 10

2 months ago

Correct Option - 2

Detailed Solution:

De Broglie wavelength

$λ = \frac{h}{m V} = \frac{h}{\sqrt[]{2 m E}} E \to K . E .$

$E = \frac{3}{2}$ KT for gas

$S o λ = \frac{h}{\sqrt[]{3 m K T}} = \frac{6.6 \times 1 0^{- 34}}{\sqrt[]{3 \times 9 \times 1 0^{- 31} \times 1.38 \times 1 0^{- 23} \times 300}}$

$\begin{array}{l} λ = 6.26 \times 1 0^{- 9} m \\ λ = 6.26 n m \end{array}$

De Broglie wavelength <math> <mrow> <mi>λ</mi> <mo>=</mo> <mfrac> <mrow> <mi>h</mi> </mrow> <mrow> <mi>m</mi> <mi>V</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>h</mi> </mrow> <mrow> <mroot> <mrow> <mn>2</mn> <mi>m</mi> <mi>E</mi> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> <mtext> </mtext> <mi>E</mi> <mo>→</mo> <mi>K</mi> <mo>.</mo> <mi>E</mi> <mo>.</mo> </mrow> </math>              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1815997971"> </o:OLEObject></xml><! [endif]-><!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1815997977"> </o:OLEObject></xml><! [endif]-> <math> <mrow> <mi>E</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>  KT for gas <math> <mrow> <mi>S</mi> <mi>o</mi> <mtext> </mtext> <mi>λ</mi> <mo>=</mo> <mfrac> <mrow> <mi>h</mi> </mrow> <mrow> <mroot> <mrow> <mn>3</mn> <mi>m</mi> <mi>K</mi> <mi>T</mi> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>6</mn> <mo>.</mo> <mn>6</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>3</mn> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mroot> <mrow> <mn>3</mn> <mo>×</mo> <mn>9</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>3</mn> <mn>1</mn> </mrow> </msup> <mo>×</mo> <mn>1</mn> <mo>.</mo> <mn>3</mn> <mn>8</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> <mn>3</mn> </mrow> </msup> <mo>×</mo> <mn>3</mn> <mn>0</mn> <mn>0</mn> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> </mrow> </math> <math> <mtable columnalign="left"> <mtr> <mtd> <mrow> <mi>λ</mi> <mo>=</mo> <mn>6</mn> <mo>.</mo> <mn>2</mn> <mn>6</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>9</mn> </mrow> </msup> <mi>m</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>λ</mi> <mo>=</mo> <mn>6</mn> <mo>.</mo> <mn>2</mn> <mn>6</mn> <mi>n</mi> <mi>m</mi> </mrow> </mtd> </mtr> </mtable> </math>              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1815997978"> </o:OLEObject></xml><! [endif]->               <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1815997979"> </o:OLEObject></xml><! [endif]->

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The temperature of an ideal gas in 3-dimensions is 300K. The corresponding de-Broglie wavelength of the electron approximately at 300 K, is:

[m_e = mass of electron = 9 × 10^-³¹kg

h = Planck constant = 6.6 × 10^-³⁴ Js

k_B = Boltzmann constant = 1.38 × 10^-²³ JK^-¹]

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The temperature of an ideal gas in 3-dimensions is 300K. The corresponding de-Broglie wavelength of the electron approximately at 300 K, is: [me = mass of electron = 9 × 10-31kg h = Planck constant = 6.6 × 10-34 Js kB = Boltzmann constant = 1.38 × 10-23 JK-1]

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The temperature of an ideal gas in 3-dimensions is 300K. The corresponding de-Broglie wavelength of the electron approximately at 300 K, is:

[m_e = mass of electron = 9 × 10^-³¹kg

h = Planck constant = 6.6 × 10^-³⁴ Js

k_B = Boltzmann constant = 1.38 × 10^-²³ JK^-¹]