a balloon filled with helium gas occupies 2.50 L at 25C and 1.00 atm. When released, it rises to an altitude where the temperature is 20C and the pressure is only 0.30 atm. Calculate the new volume of the balloon.
(P1V1/T1) = (P2V2/T2)
Remember T must be in kelvin.
To solve this problem, we can use the combined gas law equation:
(P1)(V1)/(T1) = (P2)(V2)/(T2)
Where:
P1 = initial pressure (1.00 atm)
V1 = initial volume (2.50 L)
T1 = initial temperature in Kelvin (25°C + 273.15 = 298.15 K)
P2 = final pressure (0.30 atm)
T2 = final temperature in Kelvin (20°C + 273.15 = 293.15 K)
V2 = final volume (to be calculated)
Let's plug in these values into the equation and solve for V2:
(1.00 atm)(2.50 L)/(298.15 K) = (0.30 atm)(V2)/(293.15 K)
(2.50 L)(0.30 atm)(293.15 K) = (1.00 atm)(V2)(298.15 K)
(2.50 L)(0.30 atm)(293.15 K) = (V2)(298.15 K)
(2.50 L)(0.30 atm)(293.15 K)/(298.15 K) = (V2)
(2.50 L)(0.30 atm)(0.9828) = (V2)
0.7359 L atm = V2
Thus, the new volume of the balloon at the given conditions is approximately 0.74 L.
To solve this problem, we can use the combined gas law, which relates the initial and final conditions of a gas sample. The combined gas law is expressed as:
(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, and
T1 and T2 are the initial and final temperatures.
Let's substitute the given values into the equation:
(P1 * V1) / (T1) = (P2 * V2) / (T2)
P1 = 1.00 atm
V1 = 2.50 L
T1 = 25°C + 273.15 = 298.15 K
P2 = 0.30 atm
T2 = 20°C + 273.15 = 293.15 K
Now, rearrange the equation to solve for V2:
V2 = (P1 * V1 * T2) / (P2 * T1)
Substitute the known values:
V2 = (1.00 atm * 2.50 L * 293.15 K) / (0.30 atm * 298.15 K)
Perform the calculations:
V2 = (733.875 atm⋅L⋅K) / (89.445 atm⋅K)
V2 ≈ 8.194 L
Therefore, the new volume of the balloon when it reaches the higher altitude is approximately 8.194 L.