Question

The semiconductor industry manufactures integrated circuits in large vacuum chambers where the pressure is 1.0×10−10mm of Hg.

(a) What fraction is this of atmospheric pressure? Express your answer using two significant figures.
(b) At T = 10 ∘C, how many molecules are in a cylindrical chamber 50 cm in diameter and 50 cm tall?

Answer 1

Answer:

(a): $\rm P =1.3148\times 10^{-13}}\times atmospheric\ pressure.$.

(b):  $\rm 3.35\times 10^{11}\ molecules.$

Explanation:

Part (a):

Given, the pressure in the semiconductor industry which manufactures integrated circuits in large vacuum chambers, $\rm P = 1.0\times 10^{-10}\ mm\ of\ Hg.$

We know,

1 mm of Hg = 133.322 Pa.

Therefore,

$\rm P = 1.0\times 10^{-10}\ mm\ of\ Hg= 1.0\times 10^{-10}\times 133.322\ Pa= 1.33322\times 10^{-8}\ Pa.$

1 atmospheric pressure, $\rm P_a$ = 101325 Pa.

Thus,

$\rm \dfrac{P}{P_a}=\dfrac{1.33322\times 10^{-8}\ Pa}{101325\ Pa}=1.3148\times 10^{-13}.\\\\\\\Rightarrow P =1.3148\times 10^{-13}}\ P_a.$.

Part (b):

Now,

• Temperature, $\rm T = 10\ ^\circ C=273.15+10=283.15\ K.$
• Diameter of the chamber, $\rm D=50\ cm = 0.5\ m.$
• Height of the chamber, $\rm h = 50\ cm = 0.5\ m.$

Volume of the cylindrical chamber is given by

$\rm V = \pi (Radius)^2\times Height \\=\pi \times \left ( \dfrac D2\right )^2\times h\\=\pi \times \left ( \dfrac {0.5}2\right )^2\times 0.5\\=9.82\times 10^{-2}\ m^3.$

Using the Ideal gas equation,

$\rm PV = nkT$

where,

• P = pressure of the gas.
• V = volume of the gas.
• n = number of molecules of the gas in volume V.
• k = Boltzmann constant, having value = $\rm 1.38\times 10^{-23}\ m^2 kg s^{-2} K^{-1}.$
• T = absolute temperature of the gas.

Therefore,

$\rm Number\ of\ molecules, n = \dfrac{PV}{kT}\\\\=\dfrac{ 1.33322\times 10^{-8}\times 9.82\times 10^{-2}}{1.38\times 10^{-23}\times 283.15}\\\\=3.35\times 10^{11}\ molecules.$