Compressed carbon dioxide gas is sometimes used as a coolant, but usually carbon dioxide is kept in solid blocks (known as “dry ice”). These solid blocks of carbon dioxide sublime to gas at -78°C, remaining at this temperature until they completely evaporate. There are some advantages to keeping it as a compressed gas, though – it can be kept at room temperature in self-contained and easily-filled tanks.

Let’s suppose a large tank of carbon dioxide began to leak. The gas is kept at high pressure, compressing it into a much smaller space. As the gas leaks from the container, it rapidly expands and thus cools. No matter where the leak began, the leaking gas would drift to the floor, since it is cooler than the air around it. But, as it warmed up, would the carbon dioxide stay near the floor? To answer that, we need to know the density of carbon dioxide (1.84 kg/m³) and air (1.20 kg/m³) at room temperature.

Since carbon dioxide is roughly 1.67 times denser than air, it would stay near the bottom of the room, although it would mix more as it reached room temperature. In fact, very few gasses are lighter then air when they are at the same temperature – ammonia, methane, neon, hydrogen and helium are lighter, while nitrogen is only slightly (3%) less dense.

Did you know that a carbon dioxide leak could be a fire hazard? Carbon dioxide itself is not flammable and is actually used in some fire extinguishers to suffocate flames (particularly in commercial kitchens). So where’s the danger? When such a leak occurs in an enclosed space, the cold gas sinks to the floor and pushes warmer air up to the ceiling, compressing it. This can increase the concentration of oxygen past its flashpoint so that an electrical spark near the ceiling (say in the lighting fixtures) could cause a fire, when it otherwise wouldn’t have. This is why rooms designed to store compressed gasses (even inert ones like nitrogen) are well ventilated or fitted with oxygen sensors near the floor to detect any leak quickly before it can reach dangerous levels.

Carbon actually doesn't exist as a gas at room temperature, so that one's easy--it would be solid and on the floor! Maybe you're thinking of carbon dioxide (CO₂), which is a gas at room temperature. This gas is denser (~2.0 kg/m³) than air (~1.2 kg/m³) so it will pool near the floor. You can look up the densities online to see that this is true.

An approximation that also usually works pretty well is to consider the molecular weight of the gas and compare it to nitrogen gas (N₂), since that is the biggest component in air (78%). CO₂ has a weight of 12+16+16=44 amu ("atomic mass units") for the carbon and two oxygens. Nitrogen has 14+14=28 amu. So you can see that one molecule of CO₂ is heavier than one molecule of N₂, which means it's heavier and it will sink in comparison.

All forms of pure carbon, including graphite (found in pencils) and diamond are solids at room temperature.

If a tank of carbon dioxide gas were leaking, the gas would end up near the floor. This is because CO₂ is more dense than air (about 2.0 kg/m³ for CO₂ compared to about 1.2 kg/m³ for air, at a typical temperature and pressure).

If the tank instead contained carbon monoxide gas, the gas would be found throughout the room. This is because CO gas is about the same density as air. (Of course, this isn't likely--hospitals have no reason to stock carbon monoxide, which is toxic.)

Dense fluids settle to the bottom--this is a very general rule. For example, it also explains why hot air rises: gases become less dense when heated.

That's quite an entertaining question. It would depend on how heavy the carbon gas was. Gas cylinders with carbon in them could be methane [CH₄] or carbon dioxide [CO₂] or acetylene for welding [C₂H₂] or propane [C₃H₈] or butane [C₄H₁₀], both used for heating.

If the gas is heavier than air, it will sink. If the gas is lighter than air, it will rise. That's why helium balloons rise, because helium is lighter than air.

Also, the air in the room will be mixing, so the gas will never be totally separated from the air. Butane [C₄H₁₀] and carbon dioxide are the heaviest ones listed, so they'd be lying near the floor more than the others, but the air would be moving, so they'd be mixing around.