Ideal Gas law and a hot air balloon problem

[wilsbran]

My physics problem is as following,
Estimate the average temperature of the air inside a hot air balloon. Assume that the total mass of the unfilled balloon and payload is 500kg. what is the mass of the air inside the balloon?
we are told to use a radius of 15m

Equations needed
density = m/V
ideal gas law PV=nRT
Pressure = Force over area.

This is what i Have so far:

Fnet = Fbuoyant - Fgravity
because the balloon is assumed to be in equilibrium Fnet = zero
0= density*V*g -(mass of air+mass of payload)*g
Its here that i start to get confused. I think that the density is the density of the air alone because it is the buoyant force of the heated air pushing up on the balloon so to speak.
I know that there is a relation to the gas law and from that the temperature of the balloon can be calculated...but i am stuck at this:
P= density*R*T

any suggestions/help?

 

[kuruman]

You can easily find an expression F_buoyant using Archimedes's principle. What is F_gravity? It's the weight of the unfilled balloon plus the weight of the hot air in it. If you are going to use the ideal gas law (which you should) you need to consider the pressure of the hot gas.

 

[wilsbran]

i have used the expression for the buoyant force

density*V*g = (mass of air+mass of payload)*g -------> eqn (1)

where density*V*g is the the buoyant force

i am having trouble relating it to the gas law. I know that

density = m/V

so PV=nRT can be seen as

P = density*R*T / Molar Mass

but from the force equation (eqn (1) above) i have I'm not sure how to proceed as i do not know the mass of the air.

 

[kuruman]

The mass of the air is m_air=Nm, where N = number of molecules and m = mass of one molecule. Use the ideal gas law in the form pV = NkT to replace N in the mass expressions for the gas inside and outside the balloon.

 

[paranormal]

I would approach this problem by first defining the variables and equations involved. The ideal gas law, PV=nRT, relates the pressure (P), volume (V), number of moles (n), gas constant (R), and temperature (T) of a gas. In this case, the gas in question is air.

Next, I would consider the forces acting on the hot air balloon. As you correctly stated, there is a buoyant force (Fbuoyant) acting upwards due to the heated air inside the balloon, and a gravitational force (Fgravity) acting downwards on the total mass of the balloon and payload. In order for the balloon to be in equilibrium, these two forces must be equal in magnitude.

Using the equation for density (m/V), we can calculate the density of the air inside the balloon as the buoyant force divided by the volume of the balloon. This is the density of the air alone, as you mentioned. We can then substitute this value into the ideal gas law as follows:

P = (m/V) * R * T

We know the pressure (P) is equal to the weight of the balloon and payload divided by the area of the balloon. This is because pressure is defined as force (in this case, weight) divided by area. So we can rearrange this equation to solve for the mass of the air inside the balloon (m):

m = P * V / R * T

We are given the volume of the balloon (V) and the total mass of the balloon and payload (500 kg). We can also look up the value for the gas constant (R = 8.314 J/mol·K). Therefore, we can solve for the temperature (T) by plugging in these values and solving for T.

T = P * V / (m * R)

Once we have calculated the temperature, we can use the ideal gas law to calculate the mass of the air inside the balloon. We can also use this information to estimate the average temperature of the air inside the balloon by assuming that the heated air is at a constant temperature throughout the balloon.

In summary, to solve this problem, we need to use the ideal gas law and the equations for density and pressure to calculate the temperature and mass of the air inside the hot air balloon. I hope this helps!