A meteorological balloon had a radius of 1.1 m when released at sea level at 20◦C. It ex- panded to a radius of 5 m when it had risen to its maximum altitude, where the temperature was −20◦C. What was the pressure inside the balloon at that altitude? The volume of a
sphere is V = 4 π r3 . 3
Answer in units of atm
To solve this problem, we can use the ideal gas law equation, which states:
P1V1 / T1 = P2V2 / T2
Where:
P1 = initial pressure
V1 = initial volume
T1 = initial temperature
P2 = final pressure
V2 = final volume
T2 = final temperature
We are given the initial and final radii of the balloon, as well as the initial and final temperatures. We need to find the final pressure.
Let's start by calculating the initial and final volumes of the balloon:
Initial volume (V1) = (4/3) * π * (r1^3)
Final volume (V2) = (4/3) * π * (r2^3)
where:
r1 = initial radius
r2 = final radius
Substituting the given values into the equation, we have:
Initial volume (V1) = (4/3) * π * (1.1^3)
Final volume (V2) = (4/3) * π * (5^3)
Next, we need to convert the initial and final temperatures to Kelvin.
Initial temperature (T1) = 20°C + 273.15
Final temperature (T2) = -20°C + 273.15
Now, let's substitute all the known values into the ideal gas law equation:
P1 * V1 / T1 = P2 * V2 / T2
Solving for P2, we get:
P2 = (P1 * V1 * T2) / (V2 * T1)
Substitute the known values and solve for P2 to find the pressure inside the balloon at maximum altitude. Remember to convert the final answer to atm.
Note: The units of pressure should be consistent throughout the calculation.
To find the pressure inside the balloon at its maximum altitude, we can use the ideal gas law, which states:
PV = nRT
Where:
P is the pressure inside the balloon,
V is the volume of the balloon,
n is the number of moles of gas,
R is the ideal gas constant, and
T is the temperature in Kelvin.
First, let's convert the temperatures from Celsius to Kelvin:
20°C = 20 + 273.15 = 293.15 K
-20°C = -20 + 273.15 = 253.15 K
We are given the radius at sea level (r1) and at maximum altitude (r2) of the balloon. To find the change in volume (ΔV), we can use the formula for the volume of a sphere:
ΔV = 4/3 * π * (r2^3 - r1^3)
Substituting the given values:
ΔV = 4/3 * π * (5^3 - 1.1^3) = 4/3 * π * (125 - 1.331) = 4/3 * π * 123.669 = 524.812 m^3
Next, we need to find the initial number of moles of gas inside the balloon. Assuming the balloon is filled with air, we can use the ideal gas equation PV = nRT, assuming the pressure at sea level is 1 atm, we can calculate the number of moles (n1):
(1 atm) * (4/3 * π * (1.1)^3) = n1 * (0.0821 atm*m^3/mol*K) * (293.15 K)
Simplifying:
n1 = (1 atm * 4/3 * π * (1.1)^3) / (0.0821 atm*m^3/mol*K * 293.15 K)
= 15.118 mol
Now, we can use the number of moles (n1) and the change in volume (ΔV) to find the final pressure (P2):
(P1 * V1) = (n1 * R * T1)
P2 * (V1 + ΔV) = n1 * R * T2
P2 = (n1 * R * T2) / (V1 + ΔV)
Substituting the values:
P2 = (15.118 mol * 0.0821 atm*m^3/mol*K * 253.15 K) / (4/3 * π * (1.1)^3 + 524.812 m^3)
= 1.039 atm
Therefore, the pressure inside the balloon at its maximum altitude is approximately 1.039 atm.
Ans is 1atm
The volume of a sphere is not what you have but (4/3)*pi*r^3.
Calculate volume at both heights and use P1V1 = P2V2