A hot air balloon is filled with hot air and the hot air forces the volume of the balloon to

expand until the density of the gas in the balloon is less than the density of the outside air. The
volume must be 20% larger than the balloon would be at STP in
order for the balloon to float. If you
double the temperature inside the balloon will it float?

To determine if doubling the temperature inside the balloon will make it float, we need to consider the effect of temperature on the density of air and the volume of the balloon.

According to the ideal gas law, the density of a gas is directly proportional to its pressure and inversely proportional to its temperature. Doubling the temperature without changing any other variables will result in doubling the density of the gas inside the balloon.

In order for the balloon to float, the density of the gas inside the balloon must be less than the density of the outside air. If doubling the temperature increases the density, it means the balloon will become denser and therefore will not float.

Therefore, doubling the temperature inside the balloon will not make it float.

To determine whether doubling the temperature inside the balloon will make it float, we need to understand the relationship between temperature, volume, density, and buoyancy.

The buoyancy force acting on an object submerged in a fluid (in this case, the balloon in the air) is equal to the weight of the fluid displaced by the object. For the balloon to float, it must displace an amount of air with a weight greater than its own weight.

The density of a gas is directly proportional to its pressure and inversely proportional to its temperature. The equation that relates these variables is known as the ideal gas law: PV = nRT. P represents pressure, V represents volume, n represents the number of moles of gas, R is the ideal gas constant, and T represents the temperature in Kelvin.

Given that the volume needs to be 20% larger than the volume at Standard Temperature and Pressure (STP) for the balloon to float, we can set up the equation:

(Volume at STP + 0.2 * Volume at STP) * density of outside air = Volume at STP * density of air inside the balloon

Assuming that the pressure remains constant, we can simplify the equation to:

1.2 * Volume at STP * density of outside air = Volume at STP * density of air inside the balloon

Simplifying further:

Volume at STP * (density of outside air - density of air inside the balloon) = 0.2 * Volume at STP * density of outside air

Since we want the balloon to float, density of air inside the balloon must be less than the density of outside air. Therefore, (density of outside air - density of air inside the balloon) is a positive value.

This means that the left side of the equation is greater than the right side. Thus, 0.2 * Volume at STP * density of outside air is less than 1.2 * Volume at STP * density of outside air.

Now, let's consider doubling the temperature inside the balloon. As mentioned earlier, density is inversely proportional to temperature. Doubling the temperature would decrease the density of the gas inside the balloon.

If the density of the gas inside the balloon decreases, the equation becomes:

Volume at STP * (density of outside air - decreased density of air inside the balloon) < 0.2 * Volume at STP * density of outside air

Since the decreased density of air inside the balloon is smaller, the left side of the equation will be even greater compared to the right side.

Hence, doubling the temperature inside the balloon will further decrease the density of the gas inside, making the balloon even more likely to float.