Most probable speed of Aluminium

[rashi]

Homework Statement

Figure 1 shows a type of apparatus for investigating the speed distribution f (v) of molecules in a gas. Molecules emerge into a vacuum chamber from an oven held at a constant temperature T, and are collimated by slits into a parallel beam directed towards a pair of discs rotating at a common angular speed ω, which may be varied. The discs are a fixed distance L apart (L = 262 mm) and contain narrow notches, with the second notch offset from the first by a fixed angle θ = 30° as shown. If a molecule is moving at the correct speed v, it will pass through the notches in both discs to be collected at the detector, which records the rate of arrival of molecules

The oven contains aluminium at 9.00 × 102 °C, which emerges as a monatomic beam. Calculate the speed vmp and the corresponding rotation speed ω at which the largest rate of arrival for aluminium atoms is observed.
(The molar mass of aluminium is 0.0270 kg mol−1.)

Homework Equations

Vmp= Square root 2RT/M

The Attempt at a Solution

v = ωL/θ

Can anyone please help me to taht I am doing the right thing?

 

[gneill]

Homework Statement

Figure 1 shows a type of apparatus for investigating the speed distribution f (v) of molecules in a gas. Molecules emerge into a vacuum chamber from an oven held at a constant temperature T, and are collimated by slits into a parallel beam directed towards a pair of discs rotating at a common angular speed ω, which may be varied. The discs are a fixed distance L apart (L = 262 mm) and contain narrow notches, with the second notch offset from the first by a fixed angle θ = 30° as shown. If a molecule is moving at the correct speed v, it will pass through the notches in both discs to be collected at the detector, which records the rate of arrival of molecules

The oven contains aluminium at 9.00 × 102 °C, which emerges as a monatomic beam. Calculate the speed vmp and the corresponding rotation speed ω at which the largest rate of arrival for aluminium atoms is observed.
(The molar mass of aluminium is 0.0270 kg mol−1.)

Homework Equations

Vmp= Square root 2RT/M

The Attempt at a Solution

v = ωL/θ

Can anyone please help me to taht I am doing the right thing?

Your equations look appropriate. What do you get for an answer?

 

[rashi]

I am working on it. Do I have to change degree to rad?

 

[gneill]

I am working on it. Do I have to change degree to rad?

Yes. Radians are the "natural" units for angles.

 

[rashi]

Thank you!

 

[rashi]

Why Radian intead of dgree?

Can anyone explain why we always use radian instead of degree?

 

[HallsofIvy]

In calculus, at least, the derivative (rate of change) of sin(x) is just cos(x) as long as x is in radians. If you were to use degrees it would be (180/pi) cos(x). The first formula is simpler.

Another way of looking at it (and the reason for that previous formula) is that "360 degrees in a circle" is purely arbitrary. But a circle has a "natural length", it radius. And the radian measures how many radii would fit around the part of the circle's circumference subtended by the angle, so is a "natural" measure.

 

[gneill]

Can anyone explain why we always use radian instead of degree?

Radians are intimately associated with the geometry of the circle, and it's upon the unit circle that our trigonometry is based. Length and angle relationships are natural ratios when radians are used. For example, if r is the radius of a circle, and s is some arc length measured along the circumference of the circle, then the angle that subtends s is given by s/r in radians.

Degrees are an arbitrary unit chosen because 360 happens to be a number with lots of convenient factors. In antiquity this was very handy when all calculations were performed by hand.

 

[Olliecool]

Could u please share your working.