Question

the diagrams below show two pure samples of gas in identical closed, rigid containers. each colored ball represents one gas particle. both samples have the same number of particles.
sample a
mass of each particle: 32 u
average particle speed: 1,500 m/s
sample b
mass of each particle: 32 u
average particle speed: 1,600 m/s
compare the average kinetic energies of the particles in each sample. which sample has the higher temperature?
sample a
sample b
neither, the samples have the same temperature

Explanation:

Step1: Recall the formula for kinetic energy

The formula for kinetic energy is \(KE=\frac{1}{2}mv^{2}\), where \(m\) is mass and \(v\) is velocity.

Step2: Compare the kinetic energies of the two samples

Since the mass \(m\) of each particle is the same (\(m = 32u\)) for both samples, and the average particle speed of sample \(A\) is \(v_{A}=1500m/s\) and of sample \(B\) is \(v_{B} = 1600m/s\).
We know that \(KE\propto v^{2}\) (because \(m\) is constant).
For sample \(A\), \(KE_{A}=\frac{1}{2}m\times(1500)^{2}=\frac{1}{2}m\times2250000\)
For sample \(B\), \(KE_{B}=\frac{1}{2}m\times(1600)^{2}=\frac{1}{2}m\times2560000\)
Since \(2560000>2250000\), \(KE_{B}>KE_{A}\)

Step3: Relate kinetic energy to temperature

According to the kinetic - molecular theory, the average kinetic energy of gas particles is directly proportional to the temperature of the gas (\(\overline{KE}=\frac{3}{2}kT\), where \(k\) is the Boltzmann constant and \(T\) is the absolute temperature). So, higher average kinetic energy means higher temperature.

Answer:

Sample B