Consider separate 1.0-L samples of He (g) and UF6 (g), both at 1.00 atm and containing the same number of moles. What ratio of temperatures for the two samples would produce the same root mean square velocity?
T UF6/T He= x/1.0
To solve this problem, we can use the equation for root mean square velocity, which is given by:
u = √(3RT/M)
Where:
- u is the root mean square velocity.
- R is the ideal gas constant (8.314 J/(mol·K)).
- T is the temperature in Kelvin.
- M is the molar mass of the gas.
For the ratio of root mean square velocities to be equal for both samples, the ratio of temperatures must be such that:
(√(3RT UF6/M UF6)) / (√(3RT He/M He)) = 1
Since the number of moles and pressure are the same for both samples, we can cancel them out, leaving us with:
(√(T UF6/M UF6)) / (√(T He/M He)) = 1
To solve for the ratio of temperatures (T UF6 / T He), we can square both sides of the equation:
(T UF6/M UF6) / (T He/M He) = 1
Simplifying further:
T UF6 / T He = (M UF6 / M He)
The ratio of temperatures (T UF6 / T He) is equal to the ratio of the molar masses.
So, the ratio of temperatures for the two samples would be (M UF6 / M He).
To find the ratio of temperatures that would produce the same root mean square velocity for the two samples, we can use the equation:
(T1/T2) = sqrt (m2/m1)
Where T1 and T2 are the temperatures of the two samples, m1 and m2 are the molar masses of the two gases, and the square root represents the root mean square velocity.
Given that both samples contain the same number of moles, we can assume the molar mass for UF6 is greater than the molar mass for He.
Let's assume the molar mass of He is M1 and the molar mass of UF6 is M2.
Since the number of moles in both samples is the same, we can write:
n x M1 = n x M2
Where n represents the number of moles.
Since n is cancelled out from both sides, we get:
M1 = M2
Now substituting the value of M1 = M2 in the equation (T1/T2) = sqrt (m2/m1), we get:
(T1/T2) = sqrt (M2/M1)
Further simplifying, we get:
(T1/T2) = sqrt(1)
Therefore, (T1/T2) = 1
So, the ratio of temperatures for the two samples that would produce the same root mean square velocity is 1:1, which means they should be at the same temperature.