I keep calculating the square root of 3RT/M, and am not worrying about the temperature. I keep using 238 and 235 as my M when I do calculate it for both isotopes. Then I end up getting 1.00 as my final answer. That can't possibly be correct, can it?

Question

I keep calculating the square root of 3RT/M, and am not worrying about the temperature. I keep using 238 and 235 as my M when I do calculate it for both isotopes. Then I end up getting 1.00 as my final answer. That can't possibly be correct, can it?

The question was: The two isotopes of uranium, 238U and 235U, can be separated by effusion of the corresponding UF6 gases. What is the ratio (in the form of a decimal) of the root-mean-square speed of 238UF to that of 235UF6 at constant temperature?

Answer 1

I expect you are not carrying it out to enough decimal places. The ratio of 235/238 is almost 1.00 (but not quite).

The other thing is you need to include UF6 and not just U. Sine the 3*R*T will be the same, you can get by, I think with just sqrt(1/M) vs sqrt(1/M). I went through in a really big hurry and got something like 1.004 or so but you need to do it more precisely. I worked in Oak Ridge, TN for a while (where one of the gaseous diffusion plants is located) and they have many, many separation stages (because the ratio IS so close to 1.00 as you point out).

Answer 2

To calculate the ratio of the root-mean-square (RMS) speed of 238UF6 to 235UF6, we need to use the formula for RMS speed:

Vrms = sqrt((3RT)/M)

Where:

Vrms = RMS speed
R = the ideal gas constant (8.314 J/(mol·K))
T = temperature in Kelvin
M = molar mass of the molecule

In this case, we are given that we can ignore the temperature, so we can assume it's constant.

The molar mass of 238UF6 (M1) is 238 g/mol, while the molar mass of 235UF6 (M2) is 235 g/mol.

First, let's calculate the RMS speed for 238UF6:

Vrms1 = sqrt((3RT)/M1)

And now, let's calculate the RMS speed for 235UF6:

Vrms2 = sqrt((3RT)/M2)

To find the ratio of the RMS speeds, we divide Vrms1 by Vrms2:

Ratio = Vrms1 / Vrms2

Now, let's calculate the ratio using the given molar masses:

Vrms1 = sqrt((3RT)/238)
Vrms2 = sqrt((3RT)/235)

Now, we need to input the correct temperature and calculate the ratio. However, since the question states not to worry about the temperature, we can assume a reasonable fixed temperature, such as 298 K (25 degrees Celsius).

Finally, let's plug in the values and calculate the ratio:

Vrms1 = sqrt((3 * 8.314 J/(mol·K) * 298 K) / 238)
≈ 188.18 m/s

Vrms2 = sqrt((3 * 8.314 J/(mol·K) * 298 K) / 235)
≈ 191.04 m/s

Ratio = Vrms1 / Vrms2
= 188.18 m/s / 191.04 m/s
≈ 0.985

Therefore, the ratio of the root-mean-square speed of 238UF6 to 235UF6 at constant temperature is approximately 0.985.