To determine how many molecules of carbon dioxide remain in the vessel after the pressure is reduced to 0.01 kilopascal, we can utilize the ideal gas law, which relates pressure, volume, temperature, and the number of moles of a gas. The ideal gas law is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature.
Step-by-Step Calculation
First, let's establish the initial conditions of the gas:
- Mass of CO2: 4.4 g
- Molar mass of CO2: Approximately 44 g/mol
- Initial pressure (P1): 1000 kPa
- Final pressure (P2): 0.01 kPa
Calculating Initial Moles of CO2
To find the number of moles of carbon dioxide initially present, we use the formula:
n = mass / molar mass
Substituting the values:
n = 4.4 g / 44 g/mol = 0.1 moles
Using the Ideal Gas Law
Next, we can apply the ideal gas law to find the number of moles at the final pressure. Since the temperature and volume remain constant, we can set up a ratio of the initial and final states:
(P1 * n1) = (P2 * n2)
Where:
- P1 = 1000 kPa
- n1 = 0.1 moles
- P2 = 0.01 kPa
- n2 = ? (final moles)
Rearranging the equation to solve for n2 gives:
n2 = (P1 * n1) / P2
Substituting the known values:
n2 = (1000 kPa * 0.1 moles) / 0.01 kPa
n2 = 1000 moles
Calculating the Number of Molecules
To find the number of molecules, we use Avogadro's number, which states that one mole of any substance contains approximately 6.022 x 10²³ molecules. Thus:
Number of molecules = n2 * Avogadro's number
Substituting the values:
Number of molecules = 1000 moles * 6.022 x 10²³ molecules/mole
Number of molecules ≈ 6.022 x 10²⁶ molecules
Final Thoughts
In summary, after evacuating the gas to a pressure of 0.01 kilopascal, approximately 6.022 x 10²⁶ molecules of carbon dioxide remain in the vessel. This example illustrates how changes in pressure can significantly affect the number of gas molecules present, even when temperature and volume are held constant.
