Your Book Presents All Of The Formulas You Will Need To Comp
Your Book Presents All Of The Formulas You Will Need To Complete This
Your book presents all of the formulas you will need to complete this assignment, except for the average molecular speed, shown below. 1. Let's show the formula for molecular speed at work. Provide a line-by-line solution for the molecular speed of each of the following five gases at 300K: CO, SF6, H2S, Cl2, HBr. You must show your work for each gas to get full credit (6 points for each gas). Finally, use these speeds to help place the five gases in order of increasing average molecular speed i.e., lowest to highest speeds (3 points). 2. Now let's calculate the rms speeds of CO and Cl2 gas by providing line-by-line solutions for each molecule. You must show your work for each gas to get full credit (16 points for each gas). Compare your calculated rms speeds of the CO and Cl2 molecules at 300 K by explaining any potential fundamental differences between the molecules (1 point). 3. The most probable speed formula behaves similarly to the molecular speed formula. Calculate the most probable speeds of the CO and Cl2 molecules by providing line-by-line solutions for each molecule. You must show your work for each gas to get full credit (16 points for each gas). Compare the molecular speeds of the CO and Cl2 molecules at 300 K by explaining any potential fundamental differences between the molecules (1 point). Finally, from your calculations what is the trend in similarities between rms speeds for CO and Cl2 and the most probable speeds for CO and Cl2 (1 point)?
Paper For Above instruction
The assignment requires calculating molecular speeds, root mean square (rms) speeds, and most probable speeds for five gases at a temperature of 300 K, and interpreting the differences between the molecules based on their molecular properties.
Introduction
The kinetic molecular theory provides the foundation for understanding molecular speeds of gases. It states that gases consist of particles in constant, random motion. The average behaviors of these particles—such as their molecular speeds—depend on molecular mass and temperature. Several formulas govern these speeds, including the molecular speed formula, the root mean square speed formula, and the most probable speed formula, each offering different insights into molecular motion.
---
Molecular Speed Calculation
The molecular speed (average molecular speed in this context) is given by the formula:
\[
v = \sqrt{\frac{8RT}{\pi M}}
\]
where:
- \( R \) = universal gas constant = 8.314 J/(mol·K),
- \( T \) = temperature in Kelvin,
- \( M \) = molar mass in kg/mol,
- \( \pi \) is Pi.
For calculation purposes, molar masses (g/mol) must be converted to kg/mol.
Molecular speed calculations for each gas:
1. Carbon Monoxide (CO):
- Molar mass: 28.01 g/mol = 0.02801 kg/mol
- Calculation:
\[
v_{CO} = \sqrt{\frac{8 \times 8.314 \times 300}{\pi \times 0.02801}}
\]
\[
v_{CO} = \sqrt{\frac{19954.56}{0.087999}} \approx \sqrt{226,881.5} \approx 476.86 \text{ m/s}
\]
2. Sulfur Hexafluoride (SF6):
- Molar mass: 146.06 g/mol = 0.14606 kg/mol
- Calculation:
\[
v_{SF6} = \sqrt{\frac{8 \times 8.314 \times 300}{\pi \times 0.14606}}
\]
\[
v_{SF6} = \sqrt{\frac{19954.56}{0.458} } \approx \sqrt{43,544.4} \approx 208.49 \text{ m/s}
\]
3. Hydrogen Sulfide (H2S):
- Molar mass: 34.08 g/mol = 0.03408 kg/mol
- Calculation:
\[
v_{H2S} = \sqrt{\frac{8 \times 8.314 \times 300}{\pi \times 0.03408}}
\]
\[
v_{H2S} = \sqrt{\frac{19954.56}{0.107} } \approx \sqrt{186,597} \approx 432.00 \text{ m/s}
\]
4. Chlorine (Cl2):
- Molar mass: 70.90 g/mol = 0.07090 kg/mol
- Calculation:
\[
v_{Cl_2} = \sqrt{\frac{8 \times 8.314 \times 300}{\pi \times 0.07090}}
\]
\[
v_{Cl_2} = \sqrt{\frac{19954.56}{0.2227}} \approx \sqrt{89,545} \approx 299.24 \text{ m/s}
\]
5. Hydrogen Bromide (HBr):
- Molar mass: 80.91 g/mol = 0.08091 kg/mol
- Calculation:
\[
v_{HBr} = \sqrt{\frac{8 \times 8.314 \times 300}{\pi \times 0.08091}}
\]
\[
v_{HBr} = \sqrt{\frac{19954.56}{0.254} } \approx \sqrt{78,565} \approx 280.27 \text{ m/s}
\]
Order of gases by increasing molecular speed:
\[
SF_6 (208.49)
\]
---
RMS Speed Calculations
The root mean square (rms) speed is given by:
\[
v_{rms} = \sqrt{\frac{3RT}{M}}
\]
Calculations for CO and Cl2:
1. CO:
\[
v_{rms, CO} = \sqrt{\frac{3 \times 8.314 \times 300}{0.02801}}
\]
\[
v_{rms, CO} = \sqrt{\frac{7476.6}{0.02801}} \approx \sqrt{267,007} \approx 516.87 \text{ m/s}
\]
2. Cl2:
\[
v_{rms, Cl_2} = \sqrt{\frac{3 \times 8.314 \times 300}{0.07090}}
\]
\[
v_{rms, Cl_2} = \sqrt{\frac{7476.6}{0.07090}} \approx \sqrt{105,530} \approx 324.87 \text{ m/s}
\]
Comparison and Differences:
The higher rms speed of CO relative to Cl2 stems primarily from its lower molar mass, illustrating that lighter molecules have greater average kinetic energy and speed at the same temperature.
---
Most Probable Speed Calculation
The most probable speed (\( v_{mp} \)) is specified by:
\[
v_{mp} = \sqrt{\frac{2RT}{M}}
\]
Calculations for CO and Cl2:
1. CO:
\[
v_{mp, CO} = \sqrt{\frac{2 \times 8.314 \times 300}{0.02801}} \approx \sqrt{\frac{4,989.6}{0.02801}} \approx \sqrt{178,185} \approx 421.96 \text{ m/s}
\]
2. Cl2:
\[
v_{mp, Cl_2} = \sqrt{\frac{2 \times 8.314 \times 300}{0.07090}} \approx \sqrt{\frac{4,989.6}{0.07090}} \approx \sqrt{70,368} \approx 265.46 \text{ m/s}
\]
Comparison between rms and most probable speeds:
Both speeds follow the trend that lighter molecules move faster. The rms speed exceeds the most probable speed, consistent with theoretical expectations.
---
Trends and Conclusions
The calculations reveal that for both CO and Cl2, the rms speed is higher than the most probable speed. This is because rms speed accounts for the square of velocities (highlighting the contribution of higher velocities), while the most probable speed describes the peak of the speed distribution.
Fundamentally, lighter molecules such as CO display higher average and most probable speeds compared to heavier ones like Cl2 due to their lower molar mass. These differences illustrate the inverse relationship between molar mass and molecular speed at constant temperature. The trends indicate that lighter gases tend to be more kinetically energetic, influencing their diffusion rates, effusion, and reaction kinetics in gases.
References
- Atkins, P., & de Paula, J. (2014). Physical Chemistry (10th ed.). Oxford University Press.
- Mortimer, R. (2008). Physical Chemistry. Elsevier.
- McQuarrie, D. A. (2008). Quantum Chemistry. University Science Books.
- Laidler, K. J., & Meiser, J. H. (1995). Physical Chemistry. Houghton Mifflin.
- Levine, I. N. (2014). Physical Chemistry. McGraw-Hill Education.
- Schaum's Outline of Physical Chemistry. (2015). McGraw-Hill Education.
- Moore, J. W., & Pearson, R. G. (2006). Kinetics and Mechanism. Wiley.
- Silberberg, M. S. (2012). Chemistry: The Molecular Nature of Matter and Change. McGraw-Hill Education.
- Chang, R., & Goldsby, K. (2016). Chemistry. McGraw-Hill Education.
- Atkins, P., & Jones, L. (2015). Chemical Principles. W. H. Freeman.