Textbook Question
The density of a gas of unknown molar mass was measured as a function of pressure at 0 C, as in the table that follows. (b) Why is d>P not a constant as a function of pressure?
Ch.10 - Gases
Chapter 10, Problem 113
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?
Verified step by step guidance1
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.
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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Root Mean Square Speed Formula
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.
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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.
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Related Practice
Textbook Question
You have a sample of gas at 0 C. You wish to increase therms speed by a factor of 3. To what temperature shouldthe gas be heated?
Textbook Question
Consider the following gases, all at STP: Ne, SF6, N2, CH4. (a) Which gas is most likely to depart from the assumption of the kinetic-molecular theory that says there are no attractive or repulsive forces between molecules?
Textbook Question
Consider the following gases, all at STP: Ne, SF6, N2, CH4. (d) Which one has the highest total molecular volume relative to the space occupied by the gas?
Textbook Question
Consider the following gases, all at STP: Ne, SF6, N2, CH4. (f) Which one would effuse more rapidly than N2?