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The rms speed of the atoms in a 2.0 g sample of helium gas is 700 m/s. What is the thermal energy of the gas?
21. Kinetic Theory of Ideal Gases
Average Kinetic Energy of Gases
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In a sample of gas, you pick a particle at random. The mass of the particle is 1.67 × 10-27 kg and you measure its speed to be 1600 m/s. If that particle's kinetic energy is equal to the average kinetic energy of the gas particles, what is the temperature of the sample of gas?
A
232 K
B
0.065 K
C
103.3 K
D
206.5 K
Verified step by step guidance1
Start by recalling the formula for the kinetic energy of a particle: \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the particle and \( v \) is its speed.
Substitute the given values into the kinetic energy formula: \( m = 1.67 \times 10^{-27} \text{ kg} \) and \( v = 1600 \text{ m/s} \). Calculate the kinetic energy \( KE \).
Remember that the average kinetic energy of a gas particle is also given by the formula \( KE = \frac{3}{2}kT \), where \( k \) is the Boltzmann constant \( 1.38 \times 10^{-23} \text{ J/K} \) and \( T \) is the temperature in Kelvin.
Set the kinetic energy calculated from the particle equal to the average kinetic energy formula: \( \frac{1}{2}mv^2 = \frac{3}{2}kT \).
Solve for the temperature \( T \) by rearranging the equation: \( T = \frac{mv^2}{3k} \). Substitute the known values to find the temperature of the gas sample.
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