A 3 cu ft ballon is filled with air at 90 degrees F. How many cu ft will that same ballon contain one hour after it is taken into an air conditioned room at 70 degrees F?
To calculate the final volume of the balloon after one hour in an air-conditioned room, you need to apply the ideal gas law, which states that the pressure multiplied by the volume divided by the temperature is constant.
1. Firstly, convert the temperatures from Fahrenheit to Celsius. The conversion formula is:
Celsius = (Fahrenheit - 32) * 5/9
For the initial temperature of 90 degrees F:
Celsius_initial = (90 - 32) * 5/9 = 32.2 degrees C
For the room temperature of 70 degrees F:
Celsius_final = (70 - 32) * 5/9 = 21.1 degrees C
2. Next, convert the volumes from cubic feet (cu ft) to cubic meters (cu m), as the ideal gas law requires the units to be in SI (International System).
1 cu ft is approximately equal to 0.0283 cu m.
Initial Volume = 3 cu ft * 0.0283 cu m/cu ft = 0.0849 cu m
3. Now, we can use the ideal gas law equation:
P1 * V1 / T1 = P2 * V2 / T2
Where:
P1 = initial pressure (assume constant)
V1 = initial volume
T1 = initial temperature in Kelvin
P2 = final pressure (also assume constant)
V2 = final volume (what we want to calculate)
T2 = final temperature in Kelvin
Since the problem statement does not provide pressure information and does not state whether it changes, we assume the pressure remains constant.
The final temperature in Kelvin is calculated by adding 273.15 to the final temperature in Celsius:
Kelvin_final = Celsius_final + 273.15
The ideal gas law equation can now be rearranged to solve for V2:
V2 = (P1 * V1 * T2) / (T1 * P2)
Plugging in the values:
V2 = (P1 * 0.0849 cu m * Kelvin_final) / (Celsius_initial + 273.15 * P2)
4. Since we don't know the pressure information, and it was not given in the problem, we can assume the pressure remains constant and cancels out in the equation.
Hence, the final volume, V2, will be:
V2 = (0.0849 cu m * Kelvin_final) / (Celsius_initial + 273.15)
Now you can substitute the values for Kelvin_final, Celsius_initial, and calculate V2:
V2 = (0.0849 cu m * (21.1 + 273.15)) / (32.2 + 273.15)
= (0.0849 cu m * 294.25) / 305.35
≈ 0.0818 cu m
After one hour in an air-conditioned room at 70 degrees F, the same balloon would contain approximately 0.0818 cu ft.