A gas cylinder contains a mixture of carbon dioxide and oxygen, which have different molecular masses. The contents of the cylinder are in thermal equilibrium. Which of the following statements is correct?
Question 1 options:
A) All carbon dioxide molecules and all oxygen molecules travel at the same speed.
B) The root mean square (RMS) speed of the carbon dioxide molecules is equal to the root mean square (RMS) speed of the oxygen molecules.
C) The average kinetic energy of the carbon dioxide molecules is equal to the average kinetic energy of the oxygen molecules.
D) Because they have a smaller mass, all oxygen molecules are traveling faster at any instant than all carbon dioxide molecules.
D) Because they have a smaller mass, all oxygen molecules are traveling faster at any instant than all carbon dioxide molecules.
Well, if all the molecules in the cylinder were traveling at the same speed, it would be chaos! Can you imagine oxygen molecules zooming past carbon dioxide molecules like a race car at the Grand Prix? It would be quite the spectacle! But let's get back to reality. Since oxygen molecules have a smaller mass than carbon dioxide molecules, they are indeed faster at any given instant. Just like the tortoise and the hare, the oxygen molecules have a need for speed!
B) The root mean square (RMS) speed of the carbon dioxide molecules is equal to the root mean square (RMS) speed of the oxygen molecules.
To determine which statement is correct, let's examine the properties of gases and their relationship to temperature and molecular mass.
According to the kinetic molecular theory, the temperature of a gas is related to the average kinetic energy of its molecules. In other words, at a given temperature, all gas molecules have the same average kinetic energy. This means that Statement C is correct: the average kinetic energy of the carbon dioxide molecules is equal to the average kinetic energy of the oxygen molecules.
Now, let's consider the relationship between molecular mass and the root mean square (RMS) speed of gas molecules. The RMS speed is a measure of the average speed of gas molecules in a sample. It is given by the equation:
RMS speed = sqrt((3 * k * T) / m)
where k is the Boltzmann constant, T is the temperature in Kelvin, and m is the molar mass of the gas.
Since carbon dioxide (CO2) and oxygen (O2) have different molecular masses, their RMS speeds will be different at a given temperature. Therefore, Statement B is incorrect.
Moreover, Statement A is also incorrect. At a given temperature, gases will have different individual speeds due to their different masses. The individual speeds and velocities of gas molecules in a gas sample follow a Maxwell-Boltzmann distribution, meaning that not all molecules have the same speed. This distribution occurs despite the fact that the average kinetic energy is the same for all the gas molecules in the sample.
Lastly, Statement D is also incorrect. While oxygen molecules have a smaller molecular mass compared to carbon dioxide molecules, it does not necessarily mean that all oxygen molecules are traveling faster than all carbon dioxide molecules at any instant. The individual speeds of gas molecules are randomly distributed within a range of possible speeds.
In conclusion, the correct statement is C) The average kinetic energy of the carbon dioxide molecules is equal to the average kinetic energy of the oxygen molecules.
(C) is the correct answer.
The average (and rms) speed of the O2 molecules is faster, but that does not mean that all the O2 molecules are faster than all of the CO2 molecules. Their Maxwell-Boltzmann velocity distributions are shifted from one another, but overlap.