Solved Examples of Root Mean Square Speed (v_rms) Calculations

📚 Quick Study Guide

• ⚛️ The Root Mean Square Speed ($v_{rms}$) is a way to find a typical speed of particles in a gas.
• 📏 It's calculated as the square root of the average of the squared speeds of the molecules.
• ➗ The formula is: $v_{rms} = \sqrt{\frac{3RT}{M}}$, where:
• 🌡️ R is the ideal gas constant (8.314 J/(mol·K))
• 🌡️ T is the absolute temperature (in Kelvin)
• ⚖️ M is the molar mass (in kg/mol)
• 💡 Make sure your units are consistent (SI units are preferred!)
• 🔑 Remember to convert molar mass from g/mol to kg/mol (divide by 1000).

🧪 Practice Quiz

1. Question 1: What is the RMS speed of oxygen molecules ($O_2$) at 300 K? (Molar mass of $O_2$ = 32 g/mol)
   1. A) 483.6 m/s
   2. B) 521.3 m/s
   3. C) 395.2 m/s
   4. D) 612.8 m/s
2. Question 2: If the temperature of a gas is doubled, what happens to the RMS speed?
   1. A) It doubles
   2. B) It is halved
   3. C) It increases by a factor of $\sqrt{2}$
   4. D) It decreases by a factor of $\sqrt{2}$
3. Question 3: Calculate the RMS speed of Helium atoms (He) at 4 K. (Molar mass of He = 4 g/mol)
   1. A) 25 m/s
   2. B) 158 m/s
   3. C) 49 m/s
   4. D) 316 m/s
4. Question 4: Which gas will have a higher RMS speed at the same temperature: Hydrogen ($H_2$) or Oxygen ($O_2$)?
   1. A) Hydrogen
   2. B) Oxygen
   3. C) They will be the same
   4. D) Cannot be determined
5. Question 5: What is the RMS speed of Nitrogen molecules ($N_2$) at 27°C? (Molar mass of $N_2$ = 28 g/mol)
   1. A) 412 m/s
   2. B) 515 m/s
   3. C) 621 m/s
   4. D) 398 m/s
6. Question 6: If the RMS speed of a gas is 500 m/s at 300 K, what is the RMS speed if the temperature is increased to 600 K?
   1. A) 250 m/s
   2. B) 707 m/s
   3. C) 1000 m/s
   4. D) 500 m/s
7. Question 7: A container holds two gases, A and B, at the same temperature. The molar mass of gas A is four times that of gas B. What is the ratio of the RMS speed of gas A to that of gas B ($v_{rms,A} / v_{rms,B}$)?
   1. A) 4:1
   2. B) 1:4
   3. C) 1:2
   4. D) 2:1