A weather ballon is designed to expand to a maximum radius of 20 m at its working altitude, where the air pressure is 0.030 atm and the temperature is 200 K. If the ballon is filled at atmospheric pressure and 300 K, what is its radius at liftoff?
To find the radius of the weather balloon at liftoff, we can use the ideal gas law equation:
PV = nRT
Where:
P = Pressure
V = Volume
n = Number of moles
R = Ideal Gas Constant
T = Temperature
Given that the atmospheric pressure at liftoff is 1 atm (since it is filled at atmospheric pressure) and the temperature is 300 K, we can use these values to find the initial volume.
P1 = 1 atm
T1 = 300 K
P2 = 0.030 atm
T2 = 200 K
V1/V2 = (P1/P2) * (T2/T1)
V1/V2 = (1 atm / 0.030 atm) * (200 K / 300 K)
V1/V2 = 33.3333
V1 = 33.3333 * V2
The radius of the balloon is directly proportional to the cube root of its volume. Let's call the radius at liftoff R1 and the radius at the working altitude R2.
R1 = (V1/V2)^(1/3) * R2
Since we are given that the maximum radius at the working altitude (R2) is 20 m, we can substitute this value into the equation:
R1 = (33.3333)^(1/3) * 20
Now we can calculate R1:
R1 = 3.0884 * 20
R1 ≈ 61.77 m
Therefore, the radius of the weather balloon at liftoff is approximately 61.77 meters.
To find the radius of the weather balloon at liftoff, we need to use the ideal gas law equation:
PV = nRT
Where:
P is the pressure
V is the volume
n is the number of moles of gas
R is the ideal gas constant
T is the temperature
Given:
At working altitude:
Pressure (P) = 0.030 atm
Temperature (T) = 200 K
At liftoff:
Pressure (P) = atmospheric pressure
Temperature (T) = 300 K
Let's assume that the number of moles (n) of gas remains constant throughout the process.
Now we can compare the two situations using the ideal gas law:
P1V1 = nRT1 -- Equation 1 (working altitude)
P2V2 = nRT2 -- Equation 2 (liftoff)
Since n is the same in both cases, we can set Equation 1 equal to Equation 2:
P1V1 / T1 = P2V2 / T2
Now, let's solve for V2, which is the volume of the balloon at liftoff.
V2 = (P1V1 * T2) / (P2 * T1)
Substituting the given values:
V2 = (0.030 atm * V1 * 300 K) / (atmospheric pressure * 200 K)
Since we are interested in the radius of the balloon, let's convert it from volume to radius.
The volume of a sphere is given by the formula:
V = (4/3) * π * r^3
Solving for the radius (r):
r = ((3 * V) / (4 * π))^(1/3)
Substituting V2 into the formula, we get:
r = ((3 * V2) / (4 * π))^(1/3)
Now, substitute the calculated value of V2 into the formula and solve for r to find the radius of the balloon at liftoff.