Question

In a certain region of space there are only 5 gaseous molecules per cm−3 on an average. The temperature there is 3K. The pressure of this gas is?
\[({k_B} = 1.38 \times {10^{ - 23}}mo{l^{ - 1}}{K^{ - 1}})\]

Answer 1

Hint: We know the ideal gas equation and thus how temperature, pressure, volume, and number of moles of a gas are related. Rearranging the equation in a manner where the gas constant is expressed in terms of Boltzmann constant will enable us to find the answer.

Complete step by step answer:
The ideal gas equation was formulated by combining the experimentally found laws on gases like Charles law and Boyle’s law.
Boyle’s law stated the relationship between the pressure and the volume of gas. The law is an experimental result on how the pressure of gas increases as the volume of the container decreases.
Mathematically,
\[P \propto \dfrac{1}{V}\]
On the other hand, Charles law stated the relationship between the volume and temperature of gas and showed how the volume of a gas increased with temperature.
Mathematically,
\[V \propto T\]
Combining them we obtained the ideal gas equation, which gave a single unified equation capable of establishing all the physical variables volume, pressure and temperature.
As per the law, \[\dfrac{{PV}}{T}\]is a constant for a fixed amount of gas.
For one mole of gas it was equal to \[R\]which is the universal gas constant.
Thus, the mathematical form of the law is as follows,
\[PV = nRT\]……………….. (1)
The universal gas constant R is related to the Boltzmann constant, as Boltzmann constant times Avogadro's number would result in the universal gas constant. Where the Avogadro number is \[6.022 \times {10^{23}}\].
In (1) \[n\]represents the number of moles of gas, if we are to replace it with \[\dfrac{N}{A}\]where, \[N\]is the number of molecules and \[A\]being the Avogadro’s number
The equation would become
\[PV = \dfrac{N}{A}RT\]
As, \[R = {k_B} \times A\], the equation can be rewritten as
\[PV = N{k_B}T\].
Now we have an equation relating all the quantities provided in the question (number of molecules per volume, Boltzmann constant, temperature) as well as our unknown quantity (pressure).
Now we can rearrange the variables and substitute the values.
\[P = \dfrac{{N{k_B}T}}{V}\]
\[\dfrac{N}{V}\]denotes the number of molecules per volume and is equivalent to \[5c{m^{ - 3}}\], converting to SI units \[5 \times {10^6}{m^{ - 3}}\]
Thus, \[P = 5 \times {10^6} \times 1.38 \times {10^{ - 23}} \times 3\]
Hence, pressure is \[P = 20.7 \times {10^{ - 17}}N{m^{ - 2}}\].

Note:
We can also solve it after finding the number of moles of gas and gas constant but that would make the answer unnecessarily long. Remember that the number of molecules wasn’t given instead the number of molecules per volume was given, thus volume is incorporated in the term. It is also important to check the units of values given and the proper conversion to desired units.