The length of an edge of each small cube is [tex]3.445 * 10^{-9} m[/tex].
It is given to us that -
Temperature of the ideal gas = 27.0 degree Celsius
Atmospheric pressure on the gas = 1.00 atmosphere pressure
Molecules of the gas are uniformly spread
Each molecule at the center of a small cube
We have to find out the length L of an edge of each small cube if adjacent cubes touch but don't overlap.
For solving this problem, we have to make use of the formula of Boltzmann equation for ideal gas, which can be represented as -
[tex]PV = NkT[/tex] ---- (1)
where,
P = Pressure on the gas
V = Volume of the gas
N = Number of molecules of the gas
k = Boltzmann constant
T = Temperature of the gas
According to the given information, we have -
P = 1 atm = 101325 Pa
N = 1
k = [tex]1.38 *10^{-23} J/K[/tex]
T = 27°C = 27°C + 273 = 300K
Now, substituting these values in equation (1), we have
[tex]PV = NkT\\= > 101325 * V = 1 * 1.38 *10^{-23} * 300\\= > V = \frac{414 * 10^{-23}}{101325} \\= > V = 4.0858 * 10^{-26} m^{3}[/tex]------- (2)
We know that the volume of a cube in terms of its length is given as -
[tex]V = L^{3}\\ = > L = V^{1/3}[/tex] ----- (3)
Substituting the value of V from equation (2) in equation (1), we have -
[tex]L = V^{1/3}\\= > L = (4.0858*10^{-26} )^{1/3}\\= > L = 3.445 * 10^{-9} m[/tex]
Therefore, the length of an edge of each small cube is [tex]3.445 * 10^{-9} m[/tex].
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