consider separate 1Lsamples of He and UF6 both at 1 atm and containing the same number of moles. what ratio of temps for the two samples would produce the same root mean square velocity?
tempHe/tempUF6= rms 1/2 m vhe^2/1/2 m vUF6
ratiotemps = massHe/massUF6
rms velocity = sqrt(3RT/M) where M is the molar mass. Plug in R and M for each gas, set them equal to each other, and solve for the ratio of the Ts.
To find the ratio of temperatures that would produce the same root mean square (rms) velocity for separate 1L samples of He and UF6 at 1 atm and containing the same number of moles, we need to use the concept of the ideal gas law and the equation for rms velocity.
1. Start by understanding the ideal gas law equation: PV = nRT
- P: pressure in atm (constant in this case at 1 atm)
- V: volume in liters (1L for both samples)
- n: number of moles of the gas
- R: ideal gas constant (8.314 J/(mol·K))
- T: temperature in Kelvin
2. We are given that both samples contain the same number of moles, so the 'n' value will be the same for both He and UF6.
3. Let's denote the rms velocities of He and UF6 as v1 and v2, respectively. The rms velocity can be calculated using the formula:
v = sqrt((3RT) / M)
- R: ideal gas constant (8.314 J/(mol·K))
- T: temperature in Kelvin
- M: molar mass of the gas in kg/mol
4. The molar mass of He is approximately 4 g/mol, and the molar mass of UF6 is approximately 352 g/mol.
5. We can set up the equation for the ratio of temperatures using the rms velocity formula:
v1 / v2 = sqrt(T1 / T2)
6. Squaring both sides of the equation to eliminate the square root:
(v1 / v2)^2 = T1 / T2
7. Rearranging the equation to solve for T2:
T2 = T1 / (v1 / v2)^2
Now, let's plug in the known values and solve for the ratio of temperatures:
Since both samples are at the same pressure (1 atm), their temperatures will be directly proportional to their respective rms velocities. Therefore, to find the ratio of temperatures, we need to divide the rms velocity of UF6 by the rms velocity of He.
Let's assume that the rms velocity of He (v1) is known.
1. Calculate the rms velocity of UF6 (v2) using the formula mentioned earlier, with the molar mass of UF6 (352 g/mol) and T2 as the unknown temperature.
2. Substitute the calculated v2 and known values into the equation from step 7 to find T2.
3. The ratio of temperatures, T2 / T1, would give you the answer.
Remember to convert temperature values to Kelvin before plugging them into the equations as temperature must be in Kelvin for the ideal gas law and rms velocity calculation.