At what temperature is the rms speed of helium molecules half its value at STP (0 C, 1.0 atm})?(answer in K)

Question

At what temperature is the rms speed of helium molecules half its value at STP (0 C, 1.0 atm})?(answer in K)

I think i need to use V_rms=sqrt3kT/m
k=1.38e^-23J/K
helium has 4g so i think helium molecule is 4/6.02e23
but i don't know how to proceed further

then i have to answer this also:
At what temperature is the rms speed of helium molecules twice its value at STP?(answer in K)
Thanks

Answer 1

There is an easier way to answer this problem.

Molecular speed is proportional to the square root of absolute T. To reduce speed by a factor ot 2, temperature must be reduced by a factor of 4, in this case to 273.2/4 = 68.3 K

You should be able to do the second question with the same method.

Answer 2

Your answer is correct, but i was wondering how can i use the formula to answer this question?

Answer 3

For the 2nd part i got around 1100k by playing around, but not sure if it is correct

Answer 4

The answer to the second part is 1093 K.

Since V = sqrt (3kT/m), and m and k do not change,
V2/V1 = sqrt (T2/T1, or
T2/T1 = (V2/V1)^2

There is a formula for you to use.

Answer 5

To find the temperature at which the root-mean-square (rms) speed of helium molecules is half its value at STP, you can use the formula V_rms = sqrt(3kT / m), where V_rms is the rms speed, k is the Boltzmann constant (1.38e-23 J/K), T is the temperature in Kelvin, and m is the mass of one helium molecule.

First, calculate the mass of one helium molecule using the given information: 4g is the mass of 6.02e23 helium molecules. Therefore, the mass of one helium molecule is (4g) / (6.02e23) = 6.64e-23 g.

Next, convert the mass to kilograms: 6.64e-23 g = 6.64e-26 kg.

Now, substitute the values into the formula: V_rms = sqrt(3kT / m).

Since we want to find the temperature at which the rms speed is half its value at STP, let's denote the STP temperature as T_STP.

We have two equations:
1) V_rms = sqrt(3kT / m)
2) V_rms/2 = sqrt(3kT_STP / m)

Divide equation 2 by equation 1:
(V_rms/2) / V_rms = sqrt(3kT_STP / m) / sqrt(3kT / m)

This simplifies to:
1/2 = sqrt(T_STP / T)

Square both sides:
1/4 = T_STP / T

Now, isolate T on one side:
T = T_STP * 4

Substitute in the values:
T = (0 C + 273.15) * 4 = 1092.6 K

Therefore, at a temperature of 1092.6 K, the rms speed of helium molecules will be half its value at STP.

To find the temperature at which the rms speed of helium molecules is twice its value at STP, you can follow a similar process:

Using equation 2: V_rms = sqrt(3kT_STP / m)

Let's denote the temperature at which the rms speed is twice its value at STP as T_twice:

V_rms = sqrt(3kT_twice / m)

Divide equation 2 by equation 1:
2V_rms = sqrt(3kT_twice / m) / sqrt(3kT_STP / m)

This simplifies to:
2 = sqrt(T_twice / T_STP)

Square both sides:
4 = T_twice / T_STP

Now, isolate T_twice on one side:
T_twice = T_STP * 4

Substitute in the values:
T_twice = (0 C + 273.15) * 4 = 1092.6 K

Therefore, at a temperature of 1092.6 K, the rms speed of helium molecules will be twice its value at STP.