# Ideal gas law hot air balloon problem

• lm93
In summary, the balloon would lift if the density of the surrounding air was greater then the air in the balloon.
lm93

## Homework Statement

A hot-air balloon achieves its buoyant lift by heating the air inside the balloon, which makes it less dense than the air outside. Suppose the volume of a balloon is 1900 m3 and the required lift is 2500 N (rough estimate of the weight of the equipment and passenger). Calculate the temperature of the air inside the balloon which will produce the required lift. Assume that the outside air temperature is 0°C and that air is an ideal gas under these conditions.

PV=nRT
??

## The Attempt at a Solution

I really have no idea how to start this one...I know no one is allowed to give out answers or complete solutions but I really just need help starting the problem like a little hint to get me started. I'm not asking for the answer...Thanks in advance

The 'lift' on the balloon is simply the upthrust acting on it. You can get the required mass of air in the balloon / density of air in the balloon, and the rest follows from the ideal gas law.

Ok thanks a lot; I'll try that.

I appreciate your help but I still don't know where I am going with this problem...

I tried using the buoyancy force formula to find the density of the hot air and got 0.134kg/m3 but that didn't help

Okay, well, that's a start. Do you know how to convert the ideal gas law into a form which includes density?

If not, you have the volume of the balloon. You can use the density to find the mass of air in the balloon. How can you convert that mass into something you can plug into the ideal gas law?

I found the number of moles of air but I am a bit confused as to what the pressure would be inside the balloon.

Would it be greater than, less than, or the same as the pressure outside the balloon?

Hello!

I hope this helps:
$$\Sigma F_y = F_b-F_g$$ Right?
$$F = \rho_{air-outside} V_{Ballon} g - \rho_{air-ballon} V_{Ballon} g = g V_{Ballon} \left(\rho_{air-outside}-\rho_{air-ballon}\right)$$

This eq makes sense. If the density of the surrounding air is greater then the air in the ballon, then the balloon will lift (F is positive). If the density of air of the surroundings is less then the density of air in the balloon F is negative and you sink.

Rearranging:
$$\rho_{air-ballon}=\rho_{air-outside}-\frac{F}{V_{Ballon} g}$$

For density use $$\rho=\frac{P}{R' T}$$
You might have to assume that P is taken to be at sea level and use R' for dry air.
Can you take it from there?

## 1. What is the Ideal Gas Law?

The Ideal Gas Law is a mathematical relationship between the pressure, volume, temperature, and number of moles of an ideal gas. It is represented by the formula PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is temperature.

## 2. How does the Ideal Gas Law relate to hot air balloon problems?

The Ideal Gas Law can be used to calculate the volume of gas needed to lift a hot air balloon, as well as the amount of heat needed to expand the gas to fill the balloon. This is because hot air balloons operate by heating the air inside the balloon, causing it to expand and become less dense than the surrounding air, allowing it to rise.

## 3. What factors affect the volume of gas in a hot air balloon?

The volume of gas in a hot air balloon is affected by the temperature, pressure, and number of moles of gas. As the temperature increases, the volume of gas will also increase, assuming the pressure and number of moles remain constant. Similarly, decreasing the pressure or increasing the number of moles will also increase the volume of gas.

## 4. Can the Ideal Gas Law be used for real gases in hot air balloon problems?

The Ideal Gas Law is an approximation that works well for most gases at low pressures and high temperatures. However, at high pressures and low temperatures, real gases can deviate from ideal behavior. In these cases, more complex equations, such as the Virial Equation, may need to be used.

## 5. How do you solve for the unknown variable in a hot air balloon problem using the Ideal Gas Law?

In order to solve for an unknown variable in a hot air balloon problem, you will need to rearrange the Ideal Gas Law equation to isolate the desired variable. This can be done by dividing both sides of the equation by the other variables, or by using algebraic manipulation. It is important to ensure that all units are consistent when plugging in values to the equation.

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