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## Main Question or Discussion Point

In deriving the pressure of a gas, my book states that

'if all molecules are equally likely to be travelling in any direction, the fraction whose trajectories lie in an elemental solid angle dΩ is dΩ/4π'.

This initally made sense to me, but then thinking about it, I wrote dΩ=sinθdθdφ and this means that the fraction whose trajectories lie in an elemental solid angle dΩ is sinθdθdφ/4π. This is confusing me, because the fraction whose trajectories lie in an elemental solid angle varies with θ, which I find a little contradictive given what I have quoted in the first paragraph.

One thing I have noted is that if you do the φ integral, you still have the θ dependence, and this makes sense because as θ varies the remaining annular region gets bigger.

This is just something I have never thought about before, because I then noted that the surface area of a unit sphere is

∫dA=∫dΩ=∫sinθdθdφ

and thus the area element at the top and bottom of the sphere do not contribute to the area as θ=0,π.

So, before I confuse myself even further, would anybody be able to explain why the solid angle element is varying with θ, and how this makes sense when the molecules are supposed to be equally likely to be travelling in any direction and the fraction in some elemental solid angle is dΩ/4π (which varies). Thankyou :)

'if all molecules are equally likely to be travelling in any direction, the fraction whose trajectories lie in an elemental solid angle dΩ is dΩ/4π'.

This initally made sense to me, but then thinking about it, I wrote dΩ=sinθdθdφ and this means that the fraction whose trajectories lie in an elemental solid angle dΩ is sinθdθdφ/4π. This is confusing me, because the fraction whose trajectories lie in an elemental solid angle varies with θ, which I find a little contradictive given what I have quoted in the first paragraph.

One thing I have noted is that if you do the φ integral, you still have the θ dependence, and this makes sense because as θ varies the remaining annular region gets bigger.

This is just something I have never thought about before, because I then noted that the surface area of a unit sphere is

∫dA=∫dΩ=∫sinθdθdφ

and thus the area element at the top and bottom of the sphere do not contribute to the area as θ=0,π.

So, before I confuse myself even further, would anybody be able to explain why the solid angle element is varying with θ, and how this makes sense when the molecules are supposed to be equally likely to be travelling in any direction and the fraction in some elemental solid angle is dΩ/4π (which varies). Thankyou :)