Molecular Speeds in a Laboratory Apparatus

[zferic28]

A physics student measures molecular speeds in a laboratory apparatus and concludes that the distribution of speeds is such that:
10% have a speed of 200 m/s
10% gave a speed of 250m/s
15% have a speed of 500m/s
30% have a speed of 650m/s
20% have a speed of 900m/s
15% have a speed of 1300 m/s

Calculate
a) the average speed
b) the rms speed
c)the most probable speed
Assuming that the apparatus contains and ideal gas with the molecular mass m = 50 x 10^-3 kg/mol, and the above distribution of speeds
d) determine the temperature of the gas in the apparatus

a) The precentages throw me off in this problem. I know that to find the average speed, I need to add up all the speeds^2 and divide by the total number of molecules.

[(1 x 200m/s) + (1 x 250m/s) + (1.5 x 500 m/s)+ (3.0 x 650 m/s) + (2 x 900m/s) + (1.5 x 1300m/s)]/10



b) [(1 x 200m/s)^2 + (1 x 250m/s)^2 + (1.5 x 500 m/s)^2 + (3.0 x 650 m/s)^2 + (2 x 900m/s)^2 + (1.5 x 1300m/s)^2]/10

=1.15x 10^6 m^2/s^2


The rms speed = sqrt(1.15x 10^6 m^2/s^2)

c) I really don't know what the most probable speed is or how to go about calculating it please help!

d)I know that the rms speed = sqrt(3RT/M)

I'm not too good at math ( if you didn't already realize by now) so I'm not sure how to set the equation to solve for T.

T^2 = sqrt((Vrms^2(M))/3RT)

Any help is much appreciated. Thanks in advance.

 

[HallsofIvy]

A physics student measures molecular speeds in a laboratory apparatus and concludes that the distribution of speeds is such that:
10% have a speed of 200 m/s
10% gave a speed of 250m/s
15% have a speed of 500m/s
30% have a speed of 650m/s
20% have a speed of 900m/s
15% have a speed of 1300 m/s

Calculate
a) the average speed
b) the rms speed
c)the most probable speed
Assuming that the apparatus contains and ideal gas with the molecular mass m = 50 x 10^-3 kg/mol, and the above distribution of speeds
d) determine the temperature of the gas in the apparatus

a) The precentages throw me off in this problem. I know that to find the average speed, I need to add up all the speeds^2 and divide by the total number of molecules.

[(1 x 200m/s) + (1 x 250m/s) + (1.5 x 500 m/s)+ (3.0 x 650 m/s) + (2 x 900m/s) + (1.5 x 1300m/s)]/10

No, not "speeds^2". This is just a standard average: add all the speeds and divide by the number. If the percentages are throwing you off, imagine that there are 100 molecules and use numbers of molecules instead:
10 have a speed of 200 m/s
10 gave a speed of 250m/s
15 have a speed of 500m/s
30 have a speed of 650m/s
20 have a speed of 900m/s
15 have a speed of 1300 m/s

b) [(1 x 200m/s)^2 + (1 x 250m/s)^2 + (1.5 x 500 m/s)^2 + (3.0 x 650 m/s)^2 + (2 x 900m/s)^2 + (1.5 x 1300m/s)^2]/10

=1.15x 10^6 m^2/s^2

Or you could assume 10 molecules instead! That's exactly what you did here.

The rms speed = sqrt(1.15x 10^6 m^2/s^2)

c) I really don't know what the most probable speed is or how to go about calculating it please help!

I'm not clear on what "most probable speed" means either- I suspect they are asking which speed the greatest number of molecules have. You don't have to "calculate" that- just look at your percentage table. What speed do the greatest percentage of molecules have?

d)I know that the rms speed = sqrt(3RT/M)

I'm not too good at math ( if you didn't already realize by now) so I'm not sure how to set the equation to solve for T.

T^2 = sqrt((Vrms^2(M))/3RT)

No, not T².
First get rid of the square root by squareing both sides:
(rms speed)²= 3RT/M.
Now isolate T by multiplying both sides of the equation by M and dividing both sides by 3R:
T= M(rms speed)²/3R.

Any help is much appreciated. Thanks in advance.

 

[kyle8921]

a) To find the average speed, we first need to convert the percentages into decimal form. This will give us the fraction of molecules with each speed. So, we have 0.1 molecules with a speed of 200 m/s, 0.1 molecules with a speed of 250 m/s, 0.15 molecules with a speed of 500 m/s, 0.3 molecules with a speed of 650 m/s, 0.2 molecules with a speed of 900 m/s, and 0.15 molecules with a speed of 1300 m/s.

Now, we can calculate the average speed using the formula:

Average speed = (0.1 x 200 m/s) + (0.1 x 250 m/s) + (0.15 x 500 m/s) + (0.3 x 650 m/s) + (0.2 x 900 m/s) + (0.15 x 1300 m/s)

= 65 m/s

b) The rms speed is given by the formula:

Rms speed = sqrt[(0.1 x 200 m/s)^2 + (0.1 x 250 m/s)^2 + (0.15 x 500 m/s)^2 + (0.3 x 650 m/s)^2 + (0.2 x 900 m/s)^2 + (0.15 x 1300 m/s)^2]

= 737.9 m/s

c) The most probable speed is the speed at which the maximum number of molecules have. In this case, it would be the speed with 30% of the molecules, which is 650 m/s.

d) To find the temperature of the gas, we can use the formula:

Rms speed = sqrt(3RT/M)

Rearranging this equation, we get:

T = (M x Rms speed^2)/(3R)

Substituting the given values, we get:

T = (50 x 10^-3 kg/mol x (737.9 m/s)^2)/(3 x 8.314 J/molK)

= 224.6 K

Therefore, the temperature of the gas in the apparatus is 224.6 K.