Assuming the hydrogen molecule to be a sphere with a diameter of 130 pm. use the

Question

Assuming the hydrogen molecule to be a sphere with a diameter of 130 pm. use the formula of a sphere, V_molecule = (4/3) (4/3) pir^3, to calculate the physical volume of one hydrogen molecule. Determine the percentage of the total "occupied" volume per molecule at STP (as calculated in 5(d)) that is actually occupied by the H, molecule, assuming the H_2 molecule, can be approximated by a sphere of diameter 130 pm. The difference between this percentage and one hundred percent is the percentage of "dead" or empty space in the gas sample. Determine the average center-to-center distance between nearest neighbor molecules in a sample of H_2 gas at STP. (To do this calculation, it is useful to imagine the sample of gas is a cube, which in turn is divided into smaller cubes of equal volume each with one molecule at the center of it.)

Explanation / Answer

Ans. a. Physical volume of 1 H2 molecule = 4/3 pi r3                                [radius, r = ½ diameter]

                                                                = 4/3 pi (65 pm)3

                                                                = 1150346.50998946 pm3

                                                                = 1.150 x 106 pm3

Ans. b. Volume of 1 mole H2 molecule = volume of 1 molecule x Avogadro number

                                    = 1150346.50998946 pm3 x 6.022140857 x 1023 mol-1

                                                = 6927548.71751488570536722 x 1023 pm3 mol-1

                                                = 6.92755 x 1029 pm3 mol-1

Thus, the physical volume of 1 mole of H2 molecules = 6.92755 x 1029 pm3

                                                = 6.92755 x 1029 pm3 x 10-33 L                       [1 L = 1033 pm3]

                                                = 6.92755 x 10-4 L

% physical volume occupied by H2 molecules =

(physical volume of 1 mol H2 molecules / volume of 1 mol gas at STP) x 100

= (6.92755 x 10-4 L / 22.4 L) x 100 = 0.00309 %

Ans. c. Assume, the whole 22.4 L volume of a gas at STP is occupies by 1 mole H2 moles (cubes)

Volume of a cube = a3       ; where a= length of a side

Volume of 1 mole H2 molecules (6.022140857 x 1023 molecules) at STP = 22.4 L

Volume of 1 H2 molecule (cube) = 22.4 L / 6.022140857 x 1023

                                                                = 3.7196074505568477107442722839653 x 10-23 L

                                                                = 3.7196 x 10-23 x 1033 pm3

                                                                = 3.7196 x 1011 pm3

Now, a3 = 3.7196 x 1011 pm3

Or, a = 7.191 x 103 pm

Thus, the length of 1 cube = 7.191 x 103 pm

Distance between center of two cubes = 7.191 x 103 pm

Two cubes attached together has a distance between their centers equal to their arms’ length

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