Consider the following gases, all at STP: Ne, SF6, N2, CH4. Which one has the highest root-mean-square molecular speed at a given temperature?

Name: Chemistry: The Central Science
Author: Brown
ISBN: 9780134414232

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Understand that the root-mean-square (rms) speed of a gas is given by the formula: \( v_{rms} = \sqrt{\frac{3RT}{M}} \), where \( R \) is the ideal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas in kilograms per mole.

Recognize that at Standard Temperature and Pressure (STP), the temperature \( T \) is 273.15 K. Since all gases are at the same temperature, \( T \) is constant for all gases in this problem.

Note that the rms speed is inversely proportional to the square root of the molar mass \( M \). Therefore, the gas with the smallest molar mass will have the highest rms speed.

Calculate the molar masses of the given gases: Ne (Neon) has a molar mass of approximately 20.18 g/mol, SF6 (Sulfur hexafluoride) has a molar mass of approximately 146.06 g/mol, N2 (Nitrogen) has a molar mass of approximately 28.02 g/mol, and CH4 (Methane) has a molar mass of approximately 16.04 g/mol.

Compare the molar masses: CH4 has the smallest molar mass among the given gases. Therefore, CH4 will have the highest root-mean-square molecular speed at a given temperature.

주요 개념

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Root-Mean-Square Speed

Root-mean-square (RMS) speed is a measure of the average speed of gas molecules in a sample. It is calculated using the formula v_rms = √(3RT/M), where R is the ideal gas constant, T is the temperature in Kelvin, and M is the molar mass of the gas. The RMS speed increases with temperature and decreases with increasing molar mass.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). In the context of gases, a lower molar mass results in a higher RMS speed at a given temperature, as lighter molecules move faster than heavier ones. Understanding the molar mass of the gases in question is crucial for determining which gas has the highest RMS speed.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) is defined as a temperature of 273.15 K (0°C) and a pressure of 1 atm. At STP, the behavior of ideal gases can be predicted using the ideal gas law. Knowing that all gases in the question are at STP allows for a direct comparison of their molecular speeds based solely on their molar masses.