Textbook Question
Consider the following graph. (c) For each curve, which speed is highest: the most probable speed, the root-mean-square speed, or the average speed?
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7. Gases
Maxwell-Boltzmann Distribution
Multiple Choice
Calculate the molar mass of an unknown gas if its average speed is 920 m/s at 303 K.
A
0.0105 kg/mol
B
0.0136 kg/mol
C
0.0238 kg/mol
D
0.0262 kg/mol
E
0.0281 kg/mol
Verified step by step guidance1
Start by understanding the relationship between the average speed of gas molecules and their molar mass using the formula for root-mean-square speed: \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( v_{rms} \) is the root-mean-square speed, \( R \) is the ideal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass.
Rearrange the formula to solve for molar mass \( M \): \( M = \frac{3RT}{v_{rms}^2} \). This will allow you to calculate the molar mass using the given average speed.
Substitute the known values into the equation: \( R = 8.314 \text{ J/(mol·K)} \), \( T = 303 \text{ K} \), and \( v_{rms} = 920 \text{ m/s} \).
Calculate \( v_{rms}^2 \) by squaring the average speed: \( v_{rms}^2 = (920 \text{ m/s})^2 \).
Plug the values into the rearranged formula \( M = \frac{3 \times 8.314 \times 303}{920^2} \) to find the molar mass of the gas.
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