A weather balloon is inflated to a volume of 26.2 L at a pressure of 755 mmHg and a temperature of 32.5 ∘C. The balloon rises in the atmosphere to an altitude where the pressure is 360. mmHg and the temperature is -13.5 ∘C.
Assuming the balloon can freely expand, calculate the volume of the balloon at this altitude.
To calculate the volume of the balloon at a different altitude, we can use the combined gas law, which combines Boyle's Law, Charles's Law, and Gay-Lussac's Law. The formula for the combined gas law is:
(P1 * V1) / (T1) = (P2 * V2) / (T2)
Where:
P1 and P2 are the initial and final pressures,
V1 and V2 are the initial and final volumes,
T1 and T2 are the initial and final temperatures.
We are given:
P1 = 755 mmHg
V1 = 26.2 L
T1 = 32.5°C = 32.5 + 273.15 = 305.65 K
P2 = 360 mmHg
T2 = -13.5°C = -13.5 + 273.15 = 259.65 K
Now we can substitute these values into the formula and solve for V2:
(755 mmHg * 26.2 L) / (305.65 K) = (360 mmHg * V2) / (259.65 K)
To solve for V2, we can rearrange the formula:
V2 = (755 mmHg * 26.2 L * 259.65 K) / (360 mmHg * 305.65 K)
Now we can calculate:
V2 = (19765.83 mmHg * L * K) / (110014.2 mmHg * K) ≈ 0.1795 L
Therefore, the volume of the balloon at the higher altitude is approximately 0.1795 L.