1. For nitrogen at 306 K find, a) The most probable speed b) The average speed c) The rms speed The molar mass of nitrogen is 14.0 g/mol and R=8.31 kg/(mol K) Hint: You need to convert quantities into Sl units. m/s m/s m/s
Kinetic Theory of Gas
The Kinetic Theory of gases is a classical model of gases, according to which gases are composed of molecules/particles that are in random motion. While undergoing this random motion, kinetic energy in molecules can assume random velocity across all directions. It also says that the constituent particles/molecules undergo elastic collision, which means that the total kinetic energy remains constant before and after the collision. The average kinetic energy of the particles also determines the pressure of the gas.
P-V Diagram
A P-V diagram is a very important tool of the branch of physics known as thermodynamics, which is used to analyze the working and hence the efficiency of thermodynamic engines. As the name suggests, it is used to measure the changes in pressure (P) and volume (V) corresponding to the thermodynamic system under study. The P-V diagram is used as an indicator diagram to control the given thermodynamic system.
![### Calculating Speeds of Nitrogen at 306 K
#### Problem Statement:
1. For nitrogen at 306 K, find:
- a) The most probable speed
- b) The average speed
- c) The root mean square (rms) speed
#### Given Data:
- Molar mass of nitrogen (\(M\)) = 14.0 g/mol
- Universal gas constant (\(R\)) = 8.31 J/(mol·K)
**Hint:** Remember to convert all quantities into SI units.
#### Formulas:
1. **Most Probable Speed (v_mp):**
\[
v_{mp} = \sqrt{\frac{2RT}{M}}
\]
2. **Average Speed (v_avg):**
\[
v_{avg} = \sqrt{\frac{8RT}{\pi M}}
\]
3. **Root Mean Square Speed (v_rms):**
\[
v_{rms} = \sqrt{\frac{3RT}{M}}
\]
#### Explanation:
To solve these equations, it's necessary to ensure all units are in the proper SI format:
- Convert molar mass \(M\) into kg: \( M = 14.0 \, \text{g/mol} = 0.014 \, \text{kg/mol} \)
- Use \(R = 8.31 \, \text{J/(mol·K)}\)
- Temperature \(T = 306 \, \text{K}\)
Then, apply these values to the formulas to find:
- The most probable speed (\(v_{mp}\)) in m/s
- The average speed (\(v_{avg}\)) in m/s
- The rms speed (\(v_{rms}\)) in m/s](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8f42db7f-5d63-42d4-89b9-bc5373ca5a94%2F4720a07b-62a0-4906-aacc-d448aa2c529b%2F7n5i5fe_processed.jpeg&w=3840&q=75)
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