on a warm day(92 degrees F) an air filled balloon occupies a volume of .20m^3 and is under a pressure of 20.0 lbs/in^2. if the baloon is cooled to 32 degrees F in a refigerator while its pressure is reduced to 14.7 lb/in wat is the volume on the air in the container? (assume the air behaves as an ideal gas?
Use the fact that PV/T remains constant.
V2/V1 = T2/T1 * P1/P2
The number of moles is assumed constant.
The temperatures must be absolute (Rankine or Kelvin.)
32F = 273 K = 491 R = T2
92F = 551 R = 306 K = T1
Pressures can remain in lb/in^2, since you are taking a ratio.
V2/V1 = (273/306)*(14.7/20.0)
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To solve this problem, we can use the ideal gas law equation, which relates the pressure, volume, and temperature of a gas:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles of gas
R = gas constant
T = temperature in Kelvin
First, let's convert the temperatures to Kelvin using the following formula:
T(K) = T(°C) + 273.15
Given:
Initial temperature (T1) = 92°F = (92-32)°C = 60°C = 60 + 273.15 = 333.15 K
Final temperature (T2) = 32°F = (32-32)°C = 0°C = 0 + 273.15 = 273.15 K
Also, the volume and pressure are given in SI units, so we don't need to convert them.
Now, let's calculate the initial number of moles using the ideal gas law equation. Since the volume is given in cubic meters, we need to convert the pressure from pounds per square inch (lb/in^2) to pascals (Pa). The conversion factor is 1 lb/in^2 = 6894.76 Pa.
P1 = 20 lbs/in^2 = 20 * 6894.76 Pa = 137895.2 Pa
V1 = 0.20 m^3
R = 8.314 J/(mol·K) (gas constant for ideal gases)
PV = nRT
n1 = (P1 * V1) / (R * T1)
Plug in the values:
n1 = (137895.2 Pa * 0.20 m^3) / (8.314 J/(mol·K) * 333.15 K)
Now, let's calculate the final volume using the new pressure and temperature:
P2 = 14.7 lbs/in^2 = 14.7 * 6894.76 Pa = 101.325 Pa (standard atmospheric pressure)
V2 = ?
n2 = n1 (since the number of moles doesn't change)
T2 = 273.15 K (already converted)
PV = nRT
V2 = (n2RT2) / P2
Plug in the values:
V2 = [(137895.2 Pa * 0.20 m^3) / (8.314 J/(mol·K) * 333.15 K)] * (273.15 K / 101.325 Pa)
Now, let's calculate V2:
V2 = [(137895.2 Pa * 0.20 m^3) / (8.314 J/(mol·K) * 333.15 K)] * (273.15 K / 101.325 Pa)
After performing the calculations, the final volume will be in cubic meters, which is the desired unit.
Note: Make sure to use consistent units throughout the calculations and convert units where necessary.