Question

A helium-filled weather balloon has a 0.90 m radius at liftoff where air pressure is 1.0 atm and the temperature is 298 K. When airborne, the temperature is 210 K, and its radius expands to 3.0 m. What is the pressure at the airborne location

Answer 1

The student's question about the change in pressure of a helium-filled weather balloon as it expands and cools at altitude can be answered using the ideal gas law. By comparing initial and final conditions of pressure, volume, and temperature, and using the equation P2 = P1V1T2 / (T1V2), the new pressure can be determined.

Step-by-step explanation:

The student is asking how to determine the pressure inside a helium-filled weather balloon when it rises to an altitude where the external conditions change. We are given the initial temperature, pressure, and radius of the balloon at liftoff, and the radius at its airborne location where the external temperature has decreased. To solve this problem, the ideal gas law is used in combination with the assumption that the balloon expands isotropically (uniformly in all directions). Since the internal pressure of the balloon must balance the external air pressure plus the pressure due to the tension in the balloon's material, we need to use a modified version of the ideal gas law that accounts for changes in temperature and volume to find the new pressure.

Given that the temperature and volume of the balloon change upon reaching altitude, if the volume and temperature of a gas are changed and the amount of gas (number of moles) remains constant, the ideal gas law (PV = nRT) can be rearranged to show the relationship between initial and final states:

P1V1/T1 = P2V2/T2

Where P1, V1, and T1 are the initial pressure, volume, and temperature and P2, V2, and T2 are the final pressure, volume, and temperature, respectively. Since we know all variables except for P2, we can solve for P2 by rearranging the equation to:

P2 = P1V1T2 / (T1V2)

However, it is important to note that we must convert the volumes from radius measurements to actual volumes using the formula for the volume of a sphere, V = (4/3)πr3, and we must use absolute temperatures in Kelvin.

Using this equation with the provided values (making sure to convert units where necessary), the student will be able to determine the pressure at the airborne location for the weather balloon.

Answer 2

0.019 atm

Step-by-step explanation:

Assume ideal gas, so PV/T is constant where P is pressure, V is volume and a product of radius R cubed and a constant C, T is the temperature

`(P_1V_1)/(T_1) = (P_2V_2)/(T_2)`

`P_2 = P_1(V_1)/(V_2)(T_2)/(T_1)`

`P_2 = P_1(CR_1^3)/(CR_2^3)(T_2)/(T_1)`

`P_2 = P_1\left((R_1^3)/(R_2^3)\right)^3(T_2)/(T_1)`

`P_2 = 1\left((0.9)/(3)\right)^3(210)/(298)`

`P_2 = 0.3^3*0.7 = 0.019 atm`