Get comprehensive solutions to your questions with the help of IDNLearn.com's experts. Discover reliable and timely information on any topic from our network of knowledgeable professionals.
Sagot :
To determine the temperature of the gas, we need to use the given parameters and the provided equation for translational kinetic energy. Here is a step-by-step solution:
1. Understand the Problem:
- We have a monatomic gas.
- The mass [tex]\( m \)[/tex] of the gas is 0.008 kg.
- The average speed [tex]\( v \)[/tex] of the gas molecules is 1477 m/s.
- The amount of gas [tex]\( n \)[/tex] is 2 moles.
- The gas constant [tex]\( R \)[/tex] is 8.31 J/(mol K).
2. Translational Kinetic Energy Formula:
- The translational kinetic energy of a gas is given by the formula:
[tex]\[ KE_{\text{translational}} = \frac{1}{2} m v^2 \][/tex]
- Plug in the values:
[tex]\[ KE_{\text{translational}} = \frac{1}{2} \times 0.008 \, \text{kg} \times (1477 \, \text{m/s})^2 \][/tex]
3. Calculate the Translational Kinetic Energy:
- First, square the velocity:
[tex]\[ (1477 \, \text{m/s})^2 = 2183629 \, \text{m}^2/\text{s}^2 \][/tex]
- Then multiply by the mass and by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ KE_{\text{translational}} = \frac{1}{2} \times 0.008 \, \text{kg} \times 2183629 \, \text{m}^2/\text{s}^2 \][/tex]
[tex]\[ KE_{\text{translational}} = 0.004 \times 2183629 \][/tex]
[tex]\[ KE_{\text{translational}} \approx 8726.116 \, \text{J} \][/tex]
4. Relate Kinetic Energy to Temperature:
- Use the equation:
[tex]\[ KE_{\text{translational}} = \frac{3}{2} n R T \][/tex]
- Solve for temperature [tex]\( T \)[/tex]:
[tex]\[ T = \frac{2}{3} \frac{KE_{\text{translational}}}{n R} \][/tex]
- Plug in the values:
[tex]\[ T = \frac{2}{3} \frac{8726.116 \, \text{J}}{2 \, \text{mol} \times 8.31 \, \text{J/(mol K)}} \][/tex]
5. Calculate the Temperature:
- Multiply and divide inside the brackets:
[tex]\[ T = \frac{2}{3} \frac{8726.116}{16.62} \][/tex]
[tex]\[ \frac{8726.116}{16.62} \approx 525.037063 \][/tex]
[tex]\[ T = \frac{2}{3} \times 525.037063 \approx 350.024709 \][/tex]
Thus, the temperature of the gas is approximately [tex]\( 350 \, \text{K} \)[/tex].
Therefore, the correct answer is:
A. 350 K
1. Understand the Problem:
- We have a monatomic gas.
- The mass [tex]\( m \)[/tex] of the gas is 0.008 kg.
- The average speed [tex]\( v \)[/tex] of the gas molecules is 1477 m/s.
- The amount of gas [tex]\( n \)[/tex] is 2 moles.
- The gas constant [tex]\( R \)[/tex] is 8.31 J/(mol K).
2. Translational Kinetic Energy Formula:
- The translational kinetic energy of a gas is given by the formula:
[tex]\[ KE_{\text{translational}} = \frac{1}{2} m v^2 \][/tex]
- Plug in the values:
[tex]\[ KE_{\text{translational}} = \frac{1}{2} \times 0.008 \, \text{kg} \times (1477 \, \text{m/s})^2 \][/tex]
3. Calculate the Translational Kinetic Energy:
- First, square the velocity:
[tex]\[ (1477 \, \text{m/s})^2 = 2183629 \, \text{m}^2/\text{s}^2 \][/tex]
- Then multiply by the mass and by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ KE_{\text{translational}} = \frac{1}{2} \times 0.008 \, \text{kg} \times 2183629 \, \text{m}^2/\text{s}^2 \][/tex]
[tex]\[ KE_{\text{translational}} = 0.004 \times 2183629 \][/tex]
[tex]\[ KE_{\text{translational}} \approx 8726.116 \, \text{J} \][/tex]
4. Relate Kinetic Energy to Temperature:
- Use the equation:
[tex]\[ KE_{\text{translational}} = \frac{3}{2} n R T \][/tex]
- Solve for temperature [tex]\( T \)[/tex]:
[tex]\[ T = \frac{2}{3} \frac{KE_{\text{translational}}}{n R} \][/tex]
- Plug in the values:
[tex]\[ T = \frac{2}{3} \frac{8726.116 \, \text{J}}{2 \, \text{mol} \times 8.31 \, \text{J/(mol K)}} \][/tex]
5. Calculate the Temperature:
- Multiply and divide inside the brackets:
[tex]\[ T = \frac{2}{3} \frac{8726.116}{16.62} \][/tex]
[tex]\[ \frac{8726.116}{16.62} \approx 525.037063 \][/tex]
[tex]\[ T = \frac{2}{3} \times 525.037063 \approx 350.024709 \][/tex]
Thus, the temperature of the gas is approximately [tex]\( 350 \, \text{K} \)[/tex].
Therefore, the correct answer is:
A. 350 K
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Thank you for trusting IDNLearn.com. We’re dedicated to providing accurate answers, so visit us again for more solutions.
