The Maxwell Boltzman distribution of molecular Speeds and samples of two different gases at the same temperature as shown below. Which gas has the smallest molar mass?
The Correct Answer and Explanation is:
To determine which gas has the smallest molar mass using the Maxwell-Boltzmann distribution of molecular speeds, we need to analyze the characteristics of the distribution curves for both gases at the same temperature.
The Maxwell-Boltzmann distribution describes the speed distribution of molecules in a gas. According to this distribution, the average speed of gas molecules is influenced by the molar mass of the gas. Specifically, lighter gases (those with smaller molar masses) will have a higher average molecular speed than heavier gases at the same temperature. This relationship can be described by the formula for the root-mean-square speed ((v_{rms})) of gas molecules, which is given by:
[
v_{rms} = \sqrt{\frac{3RT}{M}}
]
Where:
- (R) is the ideal gas constant,
- (T) is the absolute temperature,
- (M) is the molar mass of the gas.
From this equation, we can observe that the molar mass (M) is inversely proportional to the root-mean-square speed of the gas. This means that if one gas has a higher average speed (indicated by a peak that is shifted to the right in the Maxwell-Boltzmann distribution), it must have a lower molar mass compared to another gas with a lower average speed.
When comparing the two gas distributions, look for the position of the peak of each curve. The gas with the peak furthest to the right has a higher average speed and thus a smaller molar mass. Conversely, the gas with the peak furthest to the left will have a larger molar mass due to its lower average speed.
In conclusion, by analyzing the peaks of the Maxwell-Boltzmann distributions of the two gases, we can identify which gas has the smallest molar mass. The gas with the peak furthest to the right in the distribution graph corresponds to the gas with the smallest molar mass, demonstrating the direct relationship between molecular speed and molar mass in the context of the Maxwell-Boltzmann distribution.