At what temperature will the root-mean-square speed of oxygen molecules have the value of 640 m/s? 1 kilom has a mass of 32kg.
Well, let me put on my lab coat and juggle some numbers for you! The root-mean-square speed of gas molecules is given by the equation:
v = √(3kT/m)
Where:
- v is the root-mean-square speed of the gas molecules
- k is Boltzmann's constant (approximately 1.38 x 10^-23 J/K)
- T is the temperature in Kelvin
- m is the mass of one molecule
Now, you mentioned oxygen molecules, so we need the molar mass of oxygen. Did you mean 1 kilom (which is not a standard unit) or did you mean 1 kilogram?
To find the temperature at which the root-mean-square (RMS) speed of oxygen molecules is 640 m/s, we can use the following relationship:
RMS speed = √(3kT/m)
Where:
- RMS speed is the root-mean-square speed of the molecules
- k is the Boltzmann constant (1.38 × 10^-23 J/K),
- T is the temperature in Kelvin,
- m is the mass of the molecule.
Since you mentioned that 1 kilom (presumably kilogram) is the mass of oxygen, we'll assume that it refers to the molar mass of oxygen (O2), which is 32 g/mol or 0.032 kg/kmol.
Now, we can solve for the temperature T:
640 m/s = √(3 * 1.38 × 10^-23 J/K * T / 0.032 kg/kmol)
Squaring both sides, we get:
(640 m/s)^2 = (3 * 1.38 × 10^-23 J/K * T / 0.032 kg/kmol)
Simplifying further:
(640 m/s)^2 * 0.032 kg/kmol = 3 * 1.38 × 10^-23 J/K * T
(640^2 * 0.032) kg/mol = 4.14 × 10^-23 J/K * T
Solving for T:
T = (640^2 * 0.032) kg/mol / (4.14 × 10^-23 J/K)
Calculating the value:
T ≈ 3.93 × 10^5 K
Therefore, at a temperature of approximately 3.93 × 10^5 Kelvin, the root-mean-square speed of oxygen molecules will have a value of 640 m/s.
To find the temperature at which the root-mean-square (rms) speed of oxygen molecules is 640 m/s, we can use the formula for rms speed:
v(rms) = sqrt(3 * k * T / m)
Where:
- v(rms) is the root-mean-square speed
- k is the Boltzmann constant (1.38 * 10^-23 J/K)
- T is the temperature in Kelvin
- m is the mass of the molecule in kilograms
In this case, the molecule in question is oxygen, which has a molar mass of approximately 32 g/mol. To convert this to kilograms, we divide by 1000:
m = 32 g/mol / 1000 g/kg = 0.032 kg/mol
Now, we need to convert this to the mass of one oxygen molecule. Since 1 mole contains Avogadro's number (6.022 x 10^23) of molecules, we divide the molar mass by Avogadro's number:
mass of one oxygen molecule = 0.032 kg/mol / (6.022 x 10^23 molecules/mol) = 5.32 x 10^-26 kg/molecule
Now, we can rearrange the formula to solve for T:
T = (v(rms))^2 * m / (3 * k)
Plugging in the given values:
T = (640 m/s)^2 * (5.32 x 10^-26 kg/molecule) / (3 * 1.38 x 10^-23 J/K)
Calculating this expression will give us the temperature T in Kelvin.
You will find the formula you need at
http://chemistry.about.com/od/workedchemistryproblems/a/Calculate-Root-Mean-Square-Velocity-Of-Gas-Particles.htm
It is:
Vrms = sqrt(3RT/M)
M = 32*10^-3 kg/mol
R = ideal gas constant = 8.3145 (kg·m^2/sec^2)/K·mol
You were given the rms speed. Solve for T