At what temperature would the root-mean-square speed (thermal speed) of oxygen molecules be 89 m/s? Assume that oxygen approximates an ideal gas. The mass of one O2 molecule is 5.312 x 10-26 kg. The Boltzmann constant is 1.38 × 10-23 J/K. The answer's unit is K.
Question
At what temperature would the root-mean-square speed (thermal speed) of oxygen molecules be 89 m/s? Assume that oxygen approximates an ideal gas. The mass of one O2 molecule is 5.312 x 10-26 kg. The Boltzmann constant is 1.38 × 10-23 J/K. The answer's unit is K.
Solution
The root-mean-square speed (v_rms) of a gas is given by the formula:
v_rms = sqrt(3kT/m)
where: k is the Boltzmann constant, T is the temperature, and m is the mass of one molecule of the gas.
We can rearrange this formula to solve for T:
T = mv_rms^2 / 3k
Substituting the given values:
T = (5.312 x 10^-26 kg * (89 m/s)^2) / (3 * 1.38 x 10^-23 J/K)
T = 9.47 x 10^-24 kg*m^2/s^2 / 4.14 x 10^-23 J/K
T = 0.229 K
So, the temperature at which the root-mean-square speed of oxygen molecules would be 89 m/s is approximately 0.229 Kelvin.
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