Question

Imagine a cubic container with sides \(1 \mathrm{cm}\) in length that contains 1 atm of Ar at 298 K. How many gas-wall collisions are there per second?

Short answer

The number of gas-wall collisions per second for Argon gas in a \(1 \mathrm{cm^3}\) cubic container at 298 K and 1 atm can be calculated by: 1. Calculating the number of moles (n) using the Ideal Gas Law: \(n = \frac{PV}{RT}\) 2. Finding the number of Argon molecules (N) using Avogadro's number: \(N = n \times N_A\) 3. Calculating the average speed of Argon gas molecules (\(\bar{v}\)) using the kinetic theory of gases: \(\bar{v} = \sqrt{\frac{8k_BT}{\pi m}}\) 4. Finding the number of gas-wall collisions per second: \(\text{Collisions per second} = 6 \times \frac{N \times \bar{v}}{4V}\) By combining these steps and calculating accordingly, we can determine the number of gas-wall collisions per second for Argon gas in the container under the given conditions.

Step-by-step solution

Calculate the number of Argon gas molecules in the container

Using the ideal gas law, we have \(PV = nRT\), where: - P is the pressure (1 atm), - V is the volume (\(1 \mathrm{cm^3}\)), - n is the number of moles, - R is the gas constant (\(0.0821 \frac{\mathrm{L \cdot atm}}{\mathrm{mol \cdot K}}\)), - T is the temperature (298 K). Before we can use this equation, we need to convert the volume from cm³ to L: \(V = 1 \mathrm{cm^3} = 0.001 \mathrm{L}\) Now, solve for n: \(n = \frac{PV}{RT}\)

Find the number of molecules of Argon gas

Since we know the number of moles (n), we can find the number of molecules, N, using Avogadro's number, \(N_A\), which is: \(N_A = 6.022 \times 10^{23} \mathrm{molecules/mol}\) Hence, \(N = n \times N_A\)

Calculate the average speed of Argon gas molecules

To find the average speed, we will use the equation derived from the kinetic theory of gases: \(\bar{v} = \sqrt{\frac{8k_BT}{\pi m}}\) Where: - \(k_B\) is the Boltzmann constant (\(1.380 \times 10^{-23} \frac{\mathrm{J}}{\mathrm{K}}\)), - \(T\) is the temperature (298 K), - m is the mass of one molecule of Argon gas. We know the molar mass of Argon is approximately 40 g/mol. To find the mass of one molecule, we need to divide the molar mass by Avogadro's number and convert the mass to kg: \(m = \frac{40\, \mathrm{g/mol}}{6.022 \times 10^{23}\, \mathrm{molecules/mol}} \times \frac{1\, \mathrm{kg}}{1000\, \mathrm{g}}\) Now we can calculate \(\bar{v}\).

Find the number of gas-wall collisions per second

The number of collisions per second for each wall can be calculated using the equation: \(\text{Collisions per second} = \frac{N \times \bar{v}}{4V}\) And since we have six walls in total in the cubic container, we multiply the result by 6 to get the total number of collisions per second for all walls. Combine the steps above to find the number of gas-wall collisions per second for Argon gas in the container at the given conditions.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in the study of gases, connecting pressure (P), volume (V), temperature (T), and the amount of gas in moles (n). It is concisely expressed as
\( PV = nRT \)
where R is the universal gas constant. This law assumes that gases consist of many small particles moving in random directions with no inter-particle forces, except during collisions. In our exercise, the ideal gas law helps to determine the number of moles of argon gas in the container, which is the first step to finding the number of gas-wall collisions per second.

Avogadro's Number

Avogadro's number, commonly represented as \(N_A\), is a constant that denotes the number of constituent particles, usually atoms or molecules, in one mole of a substance. Its value is roughly
\(6.022 \times 10^{23} \)
molecules per mole. This number is pivotal in chemistry because it allows scientists to count particles by weighing. In the context of our problem, Avogadro's number enables us to convert moles of argon (found using the ideal gas law) into the actual number of argon gas molecules within our container.

Kinetic Theory of Gases

The kinetic theory of gases gives a molecular-level insight into the behavior of gases. It describes gas pressure as the result of collisions between the gas particles and the walls of their container. The theory provides equations to calculate properties like the average kinetic energy and average speed of gas particles. For example, the average speed \(\bar{v}\) of gas molecules is given by the equation
\( \bar{v} = \sqrt{\frac{8k_BT}{\pi m}} \)
where \(k_B\) is the Boltzmann constant, \(T\) is the temperature, and \(m\) is the mass of an individual gas molecule. In our case, we apply this theory to estimate the average speed of argon molecules, which is critical to determining the rate of gas-wall collisions.

Molar Mass of Argon

The molar mass of an element is the mass of one mole of that element's atoms. For argon (Ar), the molar mass is roughly 40 grams per mole. The precise determination of molar mass is essential because it allows us to relate a substance's mass to the amount in moles, facilitating calculations involving the substance's molecules or atoms. In the problem, we used the molar mass of argon to calculate the mass of a single argon molecule, which is a key step in finding the average speed of the gas molecules via the equations from the kinetic theory of gases.

Boltzmann Constant

The Boltzmann constant \(k_B\) is a fundamental physical constant that links the average kinetic energy of particles in a gas with the temperature of the gas. It is named after the Austrian physicist Ludwig Boltzmann and has a value of
\( 1.380 \times 10^{-23} \)
joules per kelvin. In thermal physics, this constant appears in numerous equations, including those from the kinetic theory of gases. During our step-by-step solution, the Boltzmann constant was vital in determining the average speed of argon gas molecules, further leading to the calculation of the gas-wall collisions per second.