Modern vacuum pumps make it easy to attain pressures of the order of ${10}^{-13}$ atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. At a pressure of $9.00\times {10}^{-14}$ atm and an ordinary temperature of $300.0$ K, how many molecules are present in a volume of $1.00$ cm³?

Verified step by step guidance

1. Start by recalling the Ideal Gas Law, which is given by the formula: $PV=nRT$, where $P$ is the pressure, $V$ is the volume, $n$ is the number of moles, $R$ is the ideal gas constant, and $T$ is the temperature.
2. Convert the volume from cm³ to liters, since the ideal gas constant $R$ is typically expressed in terms of liters. Remember that 1 cm³ is equal to 0.001 liters.
3. Use the Ideal Gas Law to solve for $n$, the number of moles. Rearrange the formula to $n=\frac{P}{V}\frac{R}{T}$. Substitute the given values: $P=9.00\; imes\; 10^\{-14\}$ atm, $V=0.001$ L, $R=0.0821$ L atm/mol K, and $T=300.0$ K.
4. Calculate the number of moles $n$ using the rearranged formula. This will give you the amount of substance in moles present in the given volume under the specified conditions.
5. Finally, convert the number of moles to the number of molecules using Avogadro's number, which is $6.022\; imes\; 10^\{23\}$ molecules/mol. Multiply the number of moles by Avogadro's number to find the total number of molecules in the 1.00 cm³ volume.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in physics that relates the pressure, volume, temperature, and number of molecules of a gas. It 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. This law assumes that the gas molecules do not interact and occupy no volume, which is a good approximation under many conditions.

Molecular Count in Gases

To find the number of molecules in a given volume of gas, we use Avogadro's number, which is 6.022 x 10^23 molecules per mole. By calculating the number of moles using the Ideal Gas Law and multiplying by Avogadro's number, we can determine the total number of molecules present. This concept is crucial for understanding the microscopic properties of gases.

Pressure Units and Conversion

Pressure is a measure of force exerted by gas molecules on the walls of a container. It is often expressed in atmospheres (atm), but can also be measured in other units like pascals (Pa) or torr. Understanding how to convert between these units is essential for solving problems involving gas laws, especially when dealing with extremely low pressures as in the given question.