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.