To calculate the root mean square (rms) velocity of the oxygen molecules in the cylinder at its maximum temperature, we can use the formula:
v_rms = sqrt((3 * k * T) / m)
Where:
- v_rms is the root mean square velocity (in m/s)
- k is Boltzmann's constant (1.38×10^-23 J/K)
- T is the temperature in Kelvin (in this case, the maximum temperature, which you mentioned as 1365 K)
- m is the molar mass of oxygen (32 g/mol or 0.032 kg/mol)
Let's plug in the values and calculate:
T = 1365 K
m = 0.032 kg/mol
k = 1.38×10^-23 J/K
v_rms = sqrt((3 * (1.38×10^-23 J/K) * (1365 K)) / (0.032 kg/mol))
Now we need to convert the given cylinder volume to the number of moles of oxygen gas.
Volume of the cylinder = 10,000 L = 10,000 * 0.001 m^3 = 10 m^3
The ideal gas law can be used to find the number of moles:
PV = nRT
Where:
- P is the pressure (in atm)
- V is the volume (in m^3)
- n is the number of moles
- R is the ideal gas constant (8.31 J/mol K)
We need to convert the pressure from atm to Pascal (Pa). 1 atm = 101325 Pa.
Pressure in Pa = 200 atm * 101325 Pa/atm = 20,265,000 Pa
Now substituting the values into the ideal gas law equation:
(20,265,000 Pa) * (10 m^3) = n * (8.31 J/mol K) * (273 K)
n = ((20,265,000 Pa) * (10 m^3)) / ((8.31 J/mol K) * (273 K))
Once we have the number of moles, we can calculate the root mean square velocity:
v_rms = sqrt((3 * (1.38×10^-23 J/K) * (1365 K)) / (n * (0.032 kg/mol)))
Now you can use a calculator to compute the value of v_rms.