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Question 41:(II) The lowest attainablepressure using the best available vacuum techniques is about 1012N/m2. At such a pressure, how many molecules are there per cm3 at 0°C?

Short Answer

Expert verified

There are 300molecules/cm3at 0C for the lowest attainable pressure.

Step by step solution

01

Given Data 

The pressure of the gas is P=1012N/m2.

The temperature of the gas is T=0C=273K

The volumeisV=1cm3.

02

Understanding of Ideal Gas Law  

The ideal gas law is the equation of state for an ideal gas. It gives the relation between the pressure (P), volume (V), and temperature (T) forn moles of the ideal gas by using the following expression:

PV=nRT

Here, R is the universal gas constant.

03

Determination of the number of molecules 

In terms of the number of molecules (N) of the gas, the ideal gas law is written as follows:

PV=NNARTPV=NkTN=PVkT

Here, k is the Boltzmann constant,NA is the number of molecules in one mole of the substance and is known as Avogadro’s number. Its value is6.02×1023.

Substitute the values into the above equation.

\(\begin{array}{c}N = \frac{{{{10}^{ - 12}}\;{\rm{N/}}{{\rm{m}}^{\rm{2}}} \times \left( {1\;{\rm{c}}{{\rm{m}}^{\rm{3}}} \times \frac{{1\;{{\rm{m}}^3}}}{{{{10}^6}\;{\rm{c}}{{\rm{m}}^{\rm{3}}}}}} \right)}}{{1.38 \times {{10}^{23}}\;{\rm{J/K}} \times 273\;{\rm{K}}}}\ = 2.65 \times

{10^2}\;{\rm{molecules/c}}{{\rm{m}}^{\rm{3}}}\ \approx 300\;{\rm{molecules/c}}{{\rm{m}}^{\rm{3}}}\end{array}\)

Thus, there are 300molecules/cm3at 0C for the lowest attainable pressure.

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