Exercise 1.15

Transcript

Exercise 1.15 One mole of an ideal gas at standard temperature and pressure occupies 22.4 L (molar volume). What is the ratio of molar volume to the atomic volume of a mole of hydrogen ? (Take the size of hydrogen molecule to be about 1 Å). Why is this ratio so large ? Given: 𝑉_(gas)=22.4 L=22.4×10^(−3) m^(3) Take the stated molecular size as an effective radius: 𝑟≈1Å=10^(−10) m Step 1: Calculate the volume of one molecule Approximating the molecule as a sphere: Volume=v =(4)/(3)𝜋𝑟^(3) =(4)/(3)𝜋(10^(−10))^(3) ≈𝟒.𝟏𝟗 × 𝟏𝟎^(−𝟑𝟎) 𝒎^(𝟑) [; ] Step 2: Calculate the summed volume of one mole of molecules One mole contains approximately 6.022 × 10^(23) molecules: 𝑉_(molecules )=𝑁_(𝐴)𝑣 =(6.022×10^(23))(4.19×10^(−30)) ≈𝟐.𝟓𝟐 × 𝟏𝟎^(−𝟔) 𝒎^(𝟑) Step 3: Calculate the ratio 𝑅=(𝑣_(𝑔𝑎𝑠))/(𝑉_(𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠)) [; =(22.4×10^(−3))/(2.52×10^(−6))] ≈𝟖.𝟗×𝟏𝟎^(𝟑) This is approximately 10^(4) as an order-of-magnitude estimate: 𝑹≃𝟏𝟎^(𝟒) Why is the ratio so large? The gas volume includes both the molecules and the spaces between them. In a gas, the intermolecular separation is much larger than the size of a molecule, so most of the volume is space between molecules. =(4)/(3)𝜋(10^(−10))^(3) ≈𝟒.𝟏𝟗 × 𝟏𝟎^(−𝟑𝟎) 𝒎^(𝟑) [; ] Step 2: Calculate the summed volume of one mole of molecules One mole contains approximately 6.022 × 10^(23) molecules: 𝑉_(molecules )=𝑁_(𝐴)𝑣 =(6.022×10^(23))(4.19×10^(−30)) ≈𝟐.𝟓𝟐 × 𝟏𝟎^(−𝟔) 𝒎^(𝟑) Step 3: Calculate the ratio 𝑅=(𝑣_(𝑔𝑎𝑠))/(𝑉_(𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠)) [; =(22.4×10^(−3))/(2.52×10^(−6))] ≈𝟖.𝟗×𝟏𝟎^(𝟑) This is approximately 10^(4) as an order-of-magnitude estimate: 𝑹≃𝟏𝟎^(𝟒) Why is the ratio so large? The gas volume includes both the molecules and the spaces between them. In a gas, the intermolecular separation is much larger than the size of a molecule, so most of the volume is space between molecules. This is approximately 10^(4) as an order-of-magnitude estimate: 𝑹≃𝟏𝟎^(𝟒) Why is the ratio so large? The gas volume includes both the molecules and the spaces between them. In a gas, the intermolecular separation is much larger than the size of a molecule, so most of the volume is space between molecules.