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For a gas of nitrogen molecules (N2), what must the temperature be if 94.7% of all the molecules have speeds less than (a) 1500 m/s? Use Table 18.2. The molar mass of N2 is 28.0 g/mol
21. Kinetic Theory of Ideal Gases
Speed Distribution of Ideal Gases
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The escape velocity from the Earth is approximately 11.2 km/s. If the mass of helium atoms is 6.64 × 10-27 kg, at what temperature would the average speed of helium atoms be equal to the escape velocity?
A
23,700 K
B
2.11 K
C
4.5 × 10-42 K
D
3.9 × 10-20 K
Verified step by step guidance1
Understand that the average speed of gas particles is related to temperature through the kinetic theory of gases. The root-mean-square speed \( v_{rms} \) is given by the formula: \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the mass of a single particle.
Set the root-mean-square speed equal to the escape velocity. In this case, \( v_{rms} = 11.2 \text{ km/s} \). Convert this speed to meters per second for consistency in units: \( 11.2 \text{ km/s} = 11200 \text{ m/s} \).
Rearrange the formula to solve for temperature \( T \): \( T = \frac{m v_{rms}^2}{3k} \).
Substitute the known values into the equation: \( m = 6.64 \times 10^{-27} \text{ kg} \), \( v_{rms} = 11200 \text{ m/s} \), and \( k = 1.38 \times 10^{-23} \text{ J/K} \) (Boltzmann constant).
Calculate the temperature \( T \) using the substituted values to find the temperature at which the average speed of helium atoms equals the escape velocity.
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