Problem 17 - Chemistry An Atoms First Approach

Question

For each of the quantities listed below, explain which of the following properties (mass of the molecule, density of the gas sample, temperature of the gas sample, size of the molecule, and number of moles of gas) must be known to calculate the quantity. a. average kinetic energy b. average number of collisions per second with other gas molecules c. average force of each impact with the wall of the container d. root mean square velocity e. average number of collisions with a given area of the container f. distance between collisions

Step-by-Step Solution

Answer

To calculate the various quantities related to gas molecules, the required properties are as follows: a. Average kinetic energy: mass of the molecule and temperature of the gas sample. b. Average number of collisions per second with other gas molecules: size of the molecule, density of the gas sample, and temperature of the gas sample. c. Average force of each impact with the wall of the container: mass of the molecule, temperature of the gas sample, and number of moles of gas. d. Root mean square velocity: mass of the molecule and temperature of the gas sample. e. Average number of collisions with a given area of the container: density of the gas sample, temperature of the gas sample, and size of the molecule. f. Distance between collisions: size of the molecule and density of the gas sample.

Step 1: a. Average kinetic energy

To calculate the average kinetic energy, we need to know the mass of the molecule and the temperature of the gas sample. The formula for average kinetic energy is given by \(KE_{avg} = \frac{3}{2}k_BT\), where \(k_B\) is the Boltzmann constant and \(T\) is the temperature.

Step 2: b. Average number of collisions per second with other gas molecules

To calculate this quantity, we need to know the size of the molecule, density of the gas sample, and temperature of the gas sample. This is because the rate of collisions depends on the probability of interaction between the molecules, which is determined by their size and concentration in the sample, as well as their average speed, which depends on the temperature.

Step 3: c. Average force of each impact with the wall of the container

To calculate the average force of each impact, we need to know the mass of the molecule, temperature of the gas sample, and number of moles of gas. The force depends on the change in momentum of the molecule upon collision, which is influenced by the mass of the molecule and its velocity. Velocity, in turn, can be related to the temperature and the number of moles.

Step 4: d. Root mean square velocity

To calculate the root mean square velocity, we need to know the mass of the molecule and the temperature of the gas sample. The formula for root mean square velocity is given by \(v_{rms} = \sqrt{\frac{3k_BT}{m}}\), where \(k_B\) is the Boltzmann constant, \(T\) is the temperature, and \(m\) is the mass of the molecule.

Step 5: e. Average number of collisions with a given area of the container

To calculate this quantity, we need to know the density of the gas sample, temperature of the gas sample, and size of the molecule. The number of collisions with a given area will depend on the concentration of molecules in the sample, as well as their average speed, which is determined by the temperature and mass of the molecule.

Step 6: f. Distance between collisions

To calculate the distance between collisions, we need to know the size of the molecule and the density of the gas sample. The distance between collisions is determined by the probability of interaction between the molecules, which depends on their size and concentration in the sample.