Complete Question
Calculate the average volume per molecule for an ideal gas at room temperature and atmospheric pressure. Then take the cube root to get an estimate of the average distance between molecules. How does this distance compare to the size of a molecule like [tex]N_2[/tex]?
Answer:
The average volume per molecule is
[tex]\frac{V}{N} = 4.09 *10^{-26} \ m^3/molecule[/tex]
The average distance between molecules
[tex]d = 3.45 *10^{-9} \ m[/tex]
The size of [tex]N_2[/tex] is 100 times smaller than the obtained value
Explanation:
From the question we can deduce that we are considering an ideal
Generally the ideal gas equation is mathematically represented as
[tex]PV = NkT[/tex]
Here T is the room temperature with value T = 300 \ K
k is the Boltzmann constant with value [tex]k = 1.38 *10^{-23} \ J/K[/tex]
P is the atmospheric pressure with value [tex]P = 1.0 *10^{5} \ N/m^2[/tex]
N is the number of molecules
Now the volume per molecule is mathematically deduced from the above equation as
[tex]\frac{V}{N} = \frac{kT}{P}[/tex]
=> [tex]\frac{V}{N} = \frac{ 1.381 *10^{-23} * 300}{ 1.0*10^{5}}[/tex]
=> [tex]\frac{V}{N} = 4.09 *10^{-26} \ m^3/molecule[/tex]
Now the distance is mathematically evaluated as
[tex]d = \sqrt{\frac{V}{N} }[/tex]
=> [tex]d = \sqrt[3]{4.09*10^{-26}}[/tex]
=> [tex]d = 3.45 *10^{-9} \ m[/tex]
Generally the size of [tex]N_2[/tex] is 115 pm which is 100 times smaller than the obtained value
Generally the size of [tex]H_2O[/tex] is [tex]95.84 \ pm[/tex] which is 10 times smaller than the obtained value
Based on the ideal gas equation, the calculated values are as follows:
The average volume per molecule is calculated using the given formula derived from the ideal gas equation:
where:
At room temperature and atmospheric pressure;
T = 300 k
P = 1.0 × 10^5 N/m^2
Substituting the values in the equation above to calculate the average volume per molecule, V/n:
V/n = 1.38 × 10^-23 × 300/1.0 × 10^5 N/m^2
V/n = 4.09 × 10^-23 m^3/molecule
Thus, the average volume per molecule is 4.09 × 10^-23 m^3/molecule
The average distance between molecules can be determined by using the formula:
[tex]d = \sqrt[3]{ \frac{v}{n} } [/tex]
Substituting the values for V/n
[tex]d = \sqrt[3]{4.09 \times {10}^{ - 23} } [/tex]
d = 3.45 × 10^-9 m or 3.45 nm
Therefore, the average distance between molecules is 3.45 nm
A molecule of N2 has an average size of 115 pm or 0.115 nm.
Comparing the two values:
3.45/0.115 = 30
Therefore, the N2 molecule is about 30 times smaller than the average distance between molecules.
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