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

The pressure in the vacuum chamber is approximately 1.32x10^-13 times the atmospheric pressure. To find the number of molecules in the cylindrical chamber, use the ideal gas equation, PV = nRT.

Step-by-step explanation:

(a) To find the fraction of atmospheric pressure, we can use the pressure equivalencies:
1 atm = 760 mmHg

So, to express the pressure of 1.0x10^-10 mmHg in terms of atm:

1.0x10^-10 mmHg × (1 atm / 760 mmHg) = 1.32x10^-13 atm

This means that the pressure in the vacuum chamber is approximately 1.32x10^-13 times the atmospheric pressure.

(b) To find the number of molecules in the cylindrical chamber, we can use the ideal gas equation:

PV = nRT

Where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. Given the diameter and height of the chamber, we can calculate the volume, and then use the ideal gas equation to find the number of molecules.

Answer 2

(a):
`\rm P =1.3148* 10^(-13)}* atmospheric\ pressure.`.

(b):
`\rm 3.35* 10^(11)\ molecules.`

Step-by-step explanation:

Part (a):

Given, the pressure in the semiconductor industry which manufactures integrated circuits in large vacuum chambers,
`\rm P = 1.0* 10^(-10)\ mm\ of\ Hg.`

We know,

1 mm of Hg = 133.322 Pa.

Therefore,

`\rm P = 1.0* 10^(-10)\ mm\ of\ Hg= 1.0* 10^(-10)* 133.322\ Pa= 1.33322* 10^(-8)\ Pa.`

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

Thus,

`\rm (P)/(P_a)=(1.33322* 10^(-8)\ Pa)/(101325\ Pa)=1.3148* 10^(-13).\\\\\\\Rightarrow P =1.3148* 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* Height \\=\pi * \left ( \frac D2\right )^2* h\\=\pi * \left ( \frac {0.5}2\right )^2* 0.5\\=9.82* 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* 10^(-23)\ m^2 kg s^(-2) K^(-1).`
• T = absolute temperature of the gas.

Therefore,

`\rm Number\ of\ molecules, n = (PV)/(kT)\\\\=( 1.33322* 10^(-8)* 9.82* 10^(-2))/(1.38* 10^(-23)* 283.15)\\\\=3.35* 10^(11)\ molecules.`