Does the density of a gas depend on the molar mass of a gas?

No it doesn't. The density of a gas depends on the temperature that is found. If it's hot the density reduces since its molecules spread, while at cold temperatures the density rises.

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The density of water is 1 g/cm cubed and in order for the pebble to sink its density has to be greater than the density of water and in order for the twig to float its density has to be less than the density of water