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Q: How do you prove that the molar heat capacity at the constant volume?
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There is nothing to prove there. The second has been DEFINED to be the unit of time in the SI.


How do you prove that volume of an atom is chiefly empty space?

The mass and volumes of electrons are not significant compared to the volume of an atom.


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How do you prove that the mass of a substance of a density is equal to the volume of water?

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How can you prove that the volume of a sphere is equal to the volume of a cylinder provided their radii are equal and the height of the cylinder is equal to the diameter of the sphere?

I'm quite sure this is impossible to prove, because the volume of a sphere is not equal to the volume of a cylinder with the same radius and height equal to the sphere's diameter. This can be shown as: Volume of sphere = (4*pi*r3) / 3. Volume of cylinder = pi*r2*h. Here, the height, h, of the cylinder = d = 2r. So, the volume of the cylinder = pi*r2*2r = 2*pi*r3, which obviously does not equal the volume of the sphere. The volume of half a sphere (with radius r) is equal to the volume of a cylinder(whose height is equal to its radius, r) minus the volume of a cone with the same height and radius. Therefore, the volume of a sphere is just double that. If you follow the nearby link, you can see a nice demonstration of that.