Hot air balloons float in the air because of the difference in density between cold and hot air. In this problem, you will estimate the minimum temperature the gas inside the balloon needs to be, for it to take off. To do this, use the following variables and make these assumptions:

The combined weight of the pilot basket together with that of the balloon fabric and other equipment is W.
The volume of the hot air inside the balloon when it is inflated is V.
The absolute temperature of the hot air at the bottom of the balloon is T_h (where T_h is greater than T_c ).
The absolute temperature of the cold air outside the balloon is T_c and its density is p_c .
The balloon is open at the bottom, so that the pressure inside and outside the balloon is the same.
As always, treat air as an ideal gas.
Use g for the magnitude of the acceleration due to gravity.

Part A
What is the density p_h of hot air inside the balloon? Assume that this density is uniform throughout the balloon.
Express the density in terms of T_h, T_c, and p_c.

p_h = p_c * (T_c / T_h)

Well, if the density of the cold air outside the balloon is p_c, and the balloon is open at the bottom, we can assume that the density inside the balloon is also p_c. Why? Because if the balloon is open at the bottom, the pressure inside and outside the balloon will be the same, and for an ideal gas, the density is directly proportional to the pressure.

So, the density of the hot air inside the balloon, p_h, would also be p_c.

To estimate the density (p_h) of the hot air inside the balloon, we can use the ideal gas law:

p_v = p_h = p_c * (T_h / T_c)

where:
p_h = density of the hot air inside the balloon
p_c = density of the cold air outside the balloon
T_h = absolute temperature of the hot air at the bottom of the balloon
T_c = absolute temperature of the cold air outside the balloon

Note that we assume the density is uniform throughout the balloon.

So, the density of the hot air inside the balloon (p_h) can be expressed as:

p_h = p_c * (T_h / T_c)

To find the density of hot air inside the balloon, we can use the ideal gas law, which states:

PV = nRT

Where:
P is the pressure of the gas,
V is the volume of the gas,
n is the number of moles of gas,
R is the ideal gas constant, and
T is the temperature of the gas in Kelvin.

In this case, the balloon is open at the bottom, so the pressure inside and outside the balloon is the same. Therefore, we can ignore the pressure term and rearrange the equation to solve for density (p):

p = (n/V) * (R * T)

Here, n/V is the number of moles of gas per unit volume, and R is the ideal gas constant.

Since we're assuming the density is uniform throughout the balloon, we can consider the entire gas inside the balloon as a single unit. Therefore, we can write:

p_h = (n_h / V) * (R * T_h)

Where:
p_h is the density of the hot air inside the balloon,
n_h is the number of moles of hot air inside the balloon, and
T_h is the absolute temperature of the hot air.

Now, we need to express the density in terms of T_h, T_c, and p_c. We know the density of the cold air outside the balloon is p_c.

The key to finding the relationship between the densities is to use the pressure-temperature relationship for ideal gases. This relationship states that the pressure of an ideal gas is directly proportional to its temperature, assuming the volume and the number of moles remain constant.

So, the density of the cold air outside the balloon, p_c, is given by:

p_c = (n_c / V) * (R * T_c)

Where:
n_c is the number of moles of cold air, and
T_c is the absolute temperature of the cold air.

Since the balloon is open at the bottom, the pressure inside and outside the balloon is the same. Therefore, we can equate the right-hand sides of the equations for the densities of hot air and cold air:

(n_h / V) * (R * T_h) = (n_c / V) * (R * T_c)

We can cancel out the V and R terms from both sides:

(n_h / p_h) * T_h = (n_c / p_c) * T_c

Finally, solving for p_h (the density of hot air), we get:

p_h = (n_h / n_c) * (p_c / T_c) * T_h

Expressing the density of hot air inside the balloon in terms of T_h, T_c, and p_c.