In the Venus atmosphere, ~50 km above the surface the air pressure and temperature are the same as here on Earth surface, and you could literally go out and walk with a swimsuit. Furthermore, the atmosphere is made of 95% CO2, meaning that air balloon would have a lifting power due to buoyancy, much
like helium in Earth's atmosphere. Thus, establishing aerostat habitats and floating cities has been envisioned.

However, since there are several gasses that are lighter than air, and the same idea (of creating floating cities) could apply to Earth.

The idea is to try to do it on Earth.

To begin, I guess one could start manufacturing relatively small spheric balloons filled with methane (CH4) or (which is cheap, and still has a lifting power comparable to the lifting power of air on Venus). To prevent methane from fires, one would add all of those balloons into another (larger) balloon filled with Nitrogen (N2), which is rather inert, and still slightly lighter than air.

Well a standard hot air balloon lifts a basket, which is rather modest to call “city”. The Hindenberg had rooms with pianos in, which is getting more like a building, but still a far cry from a “city”. How much stuff do we need before we can sensibly call it a city, how heavy will that be, and how much lifting capacity is therefore needed?

[pocmloc], a city usually has buildings and streets, meaning that probably one would have to make them for someone to recognise it as a "city".

You need sphere with a radius of only 7.5 meters filled with CH4 gas (under 1 atm pressure) to have a lifting force compensating 1000 kilograms of load.

What [j paul] said. If you build a city on a
platform and put a dome over it, the warm air
trapped inside will be enough to float it.

But you have to build your city either very very
big (more than 1000km radius, assuming a mix of
buildings and materials similar to London, and
assuming the air inside is 5°C warmer than
outside, and that the enclosing dome is a
hemisphere), or very light (low building density,
thin topsoil, not too many roads, no fat people).

A "city" with 1000 km radius would cover an area of 3,141,592 square kilometers. European Union is only 4,324,782 square kilometers, according to Wikipedia.

Maxwell, I hope you are wrong in your estimates I personally hope to own 'cloud 941', a stately home and grounds. No top soil, aero/hydroponics. Built for comfort and privacy, the dome will keep the rain off. And with an MSF hospital in the basement, so I don't loose touch with humanity.

Well, I calculated for a 10km radius city, covered
by a 10km radius hemisphere containing air at 5°C
above external temperature.

If you assume that the density of a city is about 20
tons per square metre (which would be equivalent
to a uniform 10m of concrete everywhere; or a
reasonable mix of buildings, roads and open land),
it's a long way from floating. Even a density of 1
ton per square metre (which just gives you a 1m
layer of soil everywhere - no buildings or hills)
won't float at that size.

Of course, you could build much, much lighter (a
simple floor could be 1/100th as much weight, per
square metre), but then you wouldn't really have a
"city".

Mass of city in kg = pi x r^2 x D
where r=radius in metres, D=density in kg/m^2

Lifting capacity of dome in kg= r^3 x t x 0.008
where t = temperature difference (inside-outside)
in degrees celsius

Hence, a 10km-radius city with a density of
20,000kg/m^2 has a mass of 6 x 10^12kg, and a
lifting capacity (if temperature difference=5°C) of
only 4 x 10^10kg. A 100km city has a mass of 6 x
10^14kg and a lifting capacity of 4 x 10^13kg. A
1000km city has a mass of 6 x 10^16kg and a lifting
capacity of 4 x 10^16kg, which is close to break
even.

Reducing the city density to 1ton per square
metre doesn't reduce the necessary size of the
city a lot.

If you find it easier to think in cubes instead of
hemispheres, then each ton of city needs a
column of air 1 metre by 1 metre by 50km to
support it, given a temperature differential of
5°C.

If you replaced the atmospheric nitrogen with
helium, your ton of city would only need a 1m x
1m x 1000m column to support it. But then
everyone would sound squeaky.

As I see it even if we switched from half of a balloon to a full sphere those numbers look a bit high,

at 15°C according to ISA (International Standard Atmosphere), air has a density of approximately 1.22521 kg/m3. And at 20 the same air has a density of approximately 1.2041 kg/m3. A bit over 20 grams of buoyancy for each cubic metre.

Imagine a cube of one metre sides, that weighs one kilogram, 20 grams of buoyancy isn't much use.
Now apply square cube law.
Length,---------surface weight,-------volume / lift
1 metre,--------1 kilogram,-----------20 grams
10 metres,------100 kilograms,------20 kilograms
100 metres,----10 000 kilograms,----20 000 kilograms
1 kilometre,---1000 000 kilograms,---20 000 000 kilograms.
For each time the length is increased by 10, the surface area (and therefore it's weight) increases by 100 and the volume (and therefore the buoyancy caused by that volume) increases by 1 000.
166.66 grams per square metre is very heavy for balloon fabric, but then this will need to be more durable than the average balloon. And I like simple numbers.

A 10 km balloon of this type would lift it's self plus nineteen million nine hundred thousand metric tonnes.

I have a special supply of Manilla envelope-backs
reserved for such porpoises. Your figures for air
density are about right, and if you make a few
guesstimate and assume that pi is about 3, you're in
business.

A small apology in using the cube to show the way that volume increases faster than surface. I may have inadvertently over stated my case. Cube squire law still applies regardless of the shape, but a sphere of a given diameter, will have less volume of the same length.