Question

The pressure of sodium vapour in a \(1.0 \mathrm{~L}\) container is 10 torr at \(1000^{\circ} \mathrm{C}\). How many atoms are in the container? (a) \(9.7 \times 10^{17}\) (b) \(7.6 \times 10^{19}\) (c) \(4.2 \times 10^{17}\) (d) \(9.7 \times 10^{19}\)

Short answer

There are approximately \(7.6 \times 10^{19}\) atoms in the container.

Step-by-step solution

Convert Temperature to Kelvin

Convert the temperature from Celsius to Kelvin by using the conversion formula: Kelvin = Celsius + 273.15. So for a temperature of \(1000^\circ C\), the conversion would be \(1000 + 273.15 = 1273.15 K\).

Apply Ideal Gas Law

Use the ideal gas law to determine the number of moles (\(n\)) of sodium vapour: \(PV = nRT\). The pressure (P) is given as 10 torr, which needs to be converted to atmospheres (since the gas constant R is typically given in units involving atmospheres). There are 760 torr in one atmosphere, so \(10 \text{ torr} = \frac{10}{760} \text{ atm}\). The volume (V) is 1.0 L. The gas constant (R) is 0.0821 L⋅atm/(mol⋅K). Now use these values to solve for \(n\).

Convert Pressure to Atmospheres

Convert the pressure from torr to atmospheres. \(1 \text{ atm} = 760 \text{ torr}\). Hence, \(10 \text{ torr} = \frac{10}{760} \text{ atm}\).

Calculate Moles of Gas

Now plug in the values into the ideal gas equation in atm, L, and K: \((\frac{10}{760}) \times 1 = n \times 0.0821 \times 1273.15\). Solve for \(n\) to find the moles of sodium vapor.

Calculate Number of Atoms

Use Avogadro's number (\(6.022 \times 10^{23}\) atoms/mol) to convert the number of moles of sodium vapour to the number of atoms. If \(n\) moles of gas are present, then the number of atoms is \(n \times 6.022 \times 10^{23}\) atoms.

Combine Steps and Solve for Number of Atoms

Calculate the number of atoms by combining the results from all previous steps. Make sure that the final answer has the correct number of significant figures.

Key concepts

Avogadro's Number

To fully understand the solution to the given problem, one must first grasp the concept of Avogadro's number. Avogadro's number, which is approximately \(6.022 \times 10^{23}\), represents the number of particles, usually atoms or molecules, in one mole of a substance. This vast number allows chemists to count microscopic particles in terms of moles, making calculations at the macroscopic scale possible and practical.

The reason behind the use of Avogadro's number in the solution is to convert from moles to individual atoms. Once the number of moles of sodium vapor is calculated using the ideal gas law, you then multiply it by Avogadro's number to determine the total count of atoms contained in the vapor. Without this concept, you would not be able to link the calculated moles to actual atomic counts for practical purposes.

Mole Concept in Chemistry

Breaking down the Mole Concept in Chemistry is essential in solving problems related to quantities of substances. A mole is a unit of measurement used in chemistry to express amounts of a chemical substance. One mole corresponds to the Avogadro's number of particles of the substance.

The mole concept simplifies the relationship between macroscopic measurements and the number of atoms or molecules in a sample. In relation to the exercise, understanding that moles bridge the gap between the tangible measurements taken (volume and pressure) and the number of particles is crucial. The ideal gas law is used to determine the number of moles present in a given volume of gas under specified temperature and pressure conditions. From there, the mole concept in conjunction with Avogadro's number is what allows us to transition from moles to calculating the actual number of sodium atoms present in the container.

Gas Pressure Conversion

Understanding how to perform a Gas Pressure Conversion is vital, as pressures can be expressed in various units depending on the context. In this exercise, the conversion from torr to atmospheres was necessary because the ideal gas constant \(R\) is typically given in units of L⋅atm/mol⋅K.

To convert the pressure from torr to atmospheres, you need to know the conversion factor: \(1 \text{ atm} = 760 \text{ torr}\). Therefore, you divide the pressure in torr by 760 to obtain the equivalent pressure in atmospheres. This step is critical to ensure that when you use the ideal gas law, all units are consistent. Incorrect or skipped pressure conversion can lead to erroneous results. Always remember to perform the necessary unit conversions when working with gas laws to maintain unit consistency throughout the calculation.