If a balloon containing 3000 L of gas at 39C and 99 kPa rises to an altitude where the pressure is 45.5 kPa and the temperature is 16C, the volume of the balloon under these new conditions would be calculated using the following conversion factor ratios
Remember P1V1 = P2V2 and V1/T1 = V2/T2.
Solve the first equation for V2 = P1V1/P2 and that is the same as V2 = V1 x P1/P2. So P1/P2 is the pressure ratio (factor) you want. Do the same kind of thing for T.
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To calculate the volume of the balloon under the new conditions, you can use the combined gas law equation. The combined gas law equation combines Boyle's Law, Charles's Law, and Gay-Lussac's Law into a single equation:
(P1 * V1) / (T1) = (P2 * V2) / (T2)
where:
P1 = initial pressure (99 kPa)
V1 = initial volume (3000 L)
T1 = initial temperature (39°C + 273.15 K)
P2 = final pressure (45.5 kPa)
V2 = final volume (unknown)
T2 = final temperature (16°C + 273.15 K)
Now we can rearrange the equation to solve for V2:
V2 = (P2 * V1 * T1) / (P1 * T2)
Plug in the given values:
V2 = (45.5 kPa * 3000 L * (39°C + 273.15 K)) / (99 kPa * (16°C + 273.15 K))
Now, convert temperature from Celsius to Kelvin:
V2 = (45.5 kPa * 3000 L * (312.15 K)) / (99 kPa * (289.15 K))
Solve for V2:
V2 = (45.5 kPa * 3000 L * 312.15 K) / (99 kPa * 289.15 K)
V2 ≈ 5214.82 L
Therefore, the volume of the balloon under the new conditions would be approximately 5214.82 L.
To calculate the new volume of the balloon under the given conditions, we can use the combined gas law equation:
(P1 * V1) / (T1) = (P2 * V2) / (T2)
Where:
P1 = initial pressure of the gas in the balloon
V1 = initial volume of the balloon
T1 = initial temperature of the gas in the balloon
P2 = final pressure of the gas at the new altitude
V2 = final volume of the balloon (what we want to find)
T2 = final temperature of the gas at the new altitude
Using the given information:
P1 = 99 kPa
V1 = 3000 L
T1 = 39°C = 312.15 K (converted to Kelvin)
P2 = 45.5 kPa
T2 = 16°C = 289.15 K (converted to Kelvin)
Now we can plug these values into the equation:
(99 kPa * 3000 L) / (312.15 K) = (45.5 kPa * V2) / (289.15 K)
To find V2, we'll multiply both sides of the equation by (289.15 K) and divide by (45.5 kPa):
V2 = [(99 kPa * 3000 L) * (289.15 K)] / [(312.15 K) * (45.5 kPa)]
Now we can perform the calculations:
V2 = [(99 kPa * 3000 L) * (289.15 K)] / [(312.15 K) * (45.5 kPa)]
= (88935000 kPa·L·K) / (142045.725 kPa·K)
≈ 625 L
Therefore, the volume of the balloon at the new altitude and temperature would be approximately 625 liters.