A weather balloon is inflated to a volume of 27.0 at a pressure of 730 and a temperature of 31.3. The balloon rises in the atmosphere to an altitude, where the pressure is 365 and the temperature is -14.6.
Assuming the balloon can freely expand, calculate the volume of the balloon at this altitude.
use as a ideal gas behavior
See my response below.
this is a hard one.
To solve this problem, we can use the ideal gas law, which states:
PV = nRT
Where:
- P is the pressure
- V is the volume
- n is the number of moles
- R is the gas constant (8.314 J/(mol·K))
- T is the temperature in Kelvin
To find the final volume of the balloon at the new altitude, we need to keep the number of moles constant and apply the ideal gas law equation using the initial and final conditions.
Step 1: Convert the initial and final temperatures to Kelvin:
Initial temperature: T1 = 31.3 + 273.15 = 304.45 K
Final temperature: T2 = -14.6 + 273.15 = 258.55 K
Step 2: Apply the ideal gas law for the initial and final conditions:
P1V1 = nRT1
P2V2 = nRT2
Step 3: Divide the two equations to eliminate the number of moles (n):
(V2/V1) = (P1/P2) * (T2/T1)
Step 4: Substitute the given values into the equation:
(V2/27.0) = (730/365) * (258.55/304.45)
Step 5: Solve for V2 (volume at the new altitude):
V2 = (27.0) * (730/365) * (258.55/304.45)
V2 ≈ 9.49
Therefore, the volume of the balloon at the new altitude is approximately 9.49.
To solve this problem, we can use the ideal gas law, which states that the product of pressure (P) and volume (V) is directly proportional to the product of the number of moles of gas (n) and the temperature (T). The equation is given by:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature
Since we are assuming the balloon can freely expand, the number of moles and the ideal gas constant remain constant. Therefore, we can rewrite the equation as:
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 conditions as:
P1 = 730
V1 = 27.0
T1 = 31.3
And the final conditions as:
P2 = 365
T2 = -14.6
Plugging these values into the equation, we can solve for V2, the volume at the final altitude:
(730 * 27.0) / 31.3 = (365 * V2) / (-14.6)
Cross-multiplying and solving for V2:
(730 * 27.0 * -14.6) = 365 * V2 * 31.3
V2 = (730 * 27.0 * -14.6) / (365 * 31.3)
Calculating this expression, we find:
V2 ≈ -0.0655 m³
Since volume cannot be negative, we can approximate the volume at this altitude as:
V2 ≈ 0.0655 m³
Therefore, the volume of the balloon at this altitude is approximately 0.0655 m³.