Question

Consider three one-litre flasks labeled \(A, B\) and \(C\) filled with the gases \(\mathrm{NO}, \mathrm{NO}_{2}\), and \(\mathrm{N}_{2} \mathrm{O}_{\text {; }}\) respectively, each at 1 atm and \(273 \mathrm{~K}\). In which flask do the molecules have the highest average kinetic energy? (a) Flask \(C\) (b) All are the same (c) Flask \(A\) (d) None

Short answer

All are the same.

Step-by-step solution

Understand Kinetic Energy of Gases

The average kinetic energy of gas molecules is given by the equation \(\frac{3}{2}kT\), where \(k\) is the Boltzmann constant and \(T\) is the absolute temperature. This means the average kinetic energy depends only on the temperature of the gas, not on the type of gas molecule or the volume or pressure of the gas. Since all flasks are at the same temperature, the average kinetic energy of the molecules will be the same.

Apply the Gas Law

Since all three gases are at the same temperature and pressure, and all are ideal gases, we can apply the ideal gas law, which implies that the average kinetic energy of the gas particles in each flask is the same.

Compare the Conditions of Flasks A, B, and C

Given that the conditions (temperature and pressure) of the gases in all flasks are the same and the fact that all gases behave ideally under these conditions, we can conclude that the average kinetic energy is the same in all three flasks.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in physical chemistry and physics which describes the behavior of an ideal gas. This law combines several simpler gas laws, namely Boyle's Law, Charles's Law, and Avogadro's Law, into one comprehensive formula. It is mathematically expressed as \(PV = nRT\), where:\

• \(P\) stands for the pressure of the gas,


• \(V\) represents the volume the gas occupies,


• \(n\) indicates the amount of substance (in moles) of the gas,


• \(R\) is the ideal or universal gas constant,\


• and \(T\) signifies the absolute temperature.

In essence, the Ideal Gas Law provides a clear picture of the relationship between these variables. If the temperature and the amount of a gas in a sealed container are kept constant, any increase in volume leads to a decrease in pressure and vice versa. Likewise, at a constant volume, the pressure increases with temperature.
For our exercise, since the temperature and pressure are the same for all three gas flasks, this law tells us that the gases will behave predictably and similarly regardless of the type of gas. Hence, when comparing the kinetic energies of the gases under these conditions, it's clear that they would be the same in all three cases, because they are only dependent on absolute temperature, thus supporting the conclusion of the problem.

Boltzmann Constant

The Boltzmann Constant, named after physicist Ludwig Boltzmann, is a fundamental constant that links the average kinetic energy of the particles in a gas with the temperature of the gas. It's represented by the symbol \(k\) and has a value of approximately \(1.38 \times 10^{-23} \mathrm{J/K}\). The Boltzmann Constant essentially provides a bridge between macroscopic measurements (like temperature) and microscopic behaviors (like kinetic energy of particles).
In the provided equation \(\frac{3}{2}kT\), we see that the average kinetic energy of a single molecule in a gas is proportional to the absolute temperature, with the Boltzmann Constant as the proportionality factor. This forms part of the kinetic theory of gases, which states that gas particles are in constant random motion and that the temperature of a gas is a measure of the average kinetic energy of its particles.
Understanding the role of the Boltzmann Constant in this context is crucial: it tells us that if the temperatures are equal for different gases (as in our exercise), the Boltzmann Constant ensures that their average kinetic energies will be identical as well, because it does not change from one gas to another.

Absolute Temperature

Absolute Temperature, measured in kelvins (K), is the fundamental temperature scale used in the physical sciences to describe thermodynamic phenomena. Unlike Celsius or Fahrenheit, the absolute temperature scale starts at absolute zero, which is the theoretical point at which particles have minimal vibrational motion – essentially, no thermal energy.
In terms of relating to kinetic energy, the kinetic theory of gases tells us that the absolute temperature of a gas is directly proportional to the average kinetic energy of its particles. This relationship becomes vitally important in the exercise mentioned, where all the gas samples have the same absolute temperature (\(273 K\)). This uniformity of temperature ensures that the average kinetic energies of the gas molecules are also the same, despite being different substances.
It's critical for students to recognize that the temperature, in this case, is not an arbitrary value but an absolute one, which implies that our comparison of kinetic energies across different gases is valid and meaningful. The ideal gas law, paired with an understanding of absolute temperature, allows for the conclusion that the kinetic energies in flasks \(A\), \(B\), and \(C\) are indeed identical.