Ideal gas equation and boyle's law

[moeraeleizhaj]

The mass of a hot-air balloon and its occupants is 289 kg (excluding the hot air inside the balloon). The air outside the balloon has a pressure of 1.01 x 105 Pa and a density of 1.29 kg/m3. To lift off, the air inside the balloon is heated. The volume of the heated balloon is 890 m3. The pressure of the heated air remains the same as that of the outside air. To what temperature in Kelvin must the air be heated so that the balloon just lifts off? The molecular mass of air is 29 u.


Hi, can anyone help me with this question please?
I think it has something to do with the the ideal gas equation (PV=nRT)?
and boyle's law. Can anyone help me start this question

 

[alphysicist]

Hi moeraeleizhaj,

The mass of a hot-air balloon and its occupants is 289 kg (excluding the hot air inside the balloon). The air outside the balloon has a pressure of 1.01 x 105 Pa and a density of 1.29 kg/m3. To lift off, the air inside the balloon is heated. The volume of the heated balloon is 890 m3. The pressure of the heated air remains the same as that of the outside air. To what temperature in Kelvin must the air be heated so that the balloon just lifts off? The molecular mass of air is 29 u.


Hi, can anyone help me with this question please?
I think it has something to do with the the ideal gas equation (PV=nRT)?

Yes, you will need the ideal gas law (unless they want you to use something more accurate).

and boyle's law. Can anyone help me start this question

I don't think boyle's law will help much here, since the appropriate volume is given and the pressure is constant. This problem is about the buoyant force due to hot air. What is the formula for the buoyant force, and what does it need to equal for the balloon to just rise? What answer do you get?

 

[thomate1]

?

Sure, I can help you with this question. First, let's start by understanding the ideal gas equation and Boyle's law. The ideal gas equation, PV=nRT, relates the pressure (P), volume (V), amount of gas (n), and temperature (T) of an ideal gas. Boyle's law states that at a constant temperature, the pressure and volume of a gas are inversely proportional.

In this scenario, we can use the ideal gas equation to calculate the temperature needed for the balloon to lift off. First, we need to find the amount of gas (n) inside the balloon. We can do this by subtracting the mass of the balloon and occupants (289 kg) from the total mass of the balloon and heated air (289 kg + 890 m3 x 1.29 kg/m3 = 1160.1 kg). This gives us a mass of 871.1 kg of air inside the balloon.

Next, we can use Boyle's law to set up an equation with the outside pressure and the pressure of the heated air, which we know is the same. This can be written as P1V1 = P2V2, where P1 is the outside pressure (1.01 x 105 Pa) and V1 is the volume of the balloon (890 m3). P2 and V2 represent the pressure and volume of the heated air, respectively.

Now, we can plug in the values we know into this equation and solve for the unknown temperature (T). This gives us T = (P1V1)/(nR), where R is the ideal gas constant (8.314 J/mol·K). Plugging in the values, we get T = (1.01 x 105 Pa x 890 m3)/(871.1 kg x 8.314 J/mol·K) = 1096.5 K.

Therefore, the air inside the balloon needs to be heated to a temperature of 1096.5 K (823.4 °C or 1523.1 °F) in order for the balloon to just lift off. I hope this helps and gives you a starting point for solving the question!