A helium tank has a volume of 8.0 L. It contains enough helium to fill 23 circular balloons with a radius of 15 cm at a temperature of 25.0 C (room temperature). Assume that the pressure inside the balloons is 2.0 atm. What volume would all the gas in the tank occupy at 2.0 atm and a temperature of 25.0 oC (ie. when it is put into the balloons).
PV=kT
since P and T are constant, so is V. That is,
V = (23 * 4/3 pi * 15^3)
To find the volume of gas in the tank at 2.0 atm and a temperature of 25.0 °C, we can use the combined gas law. The combined gas law states that for a given amount of gas, the pressure, volume, and temperature are related by the equation:
(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 (expressed in Kelvin)
In this case, the initial conditions are the volume of the helium tank (V₁ = 8.0 L), the temperature (T₁ = 25.0 °C = 298.15 K), and the pressure (P₁ = 1 atm).
The final conditions are the volume of the gas in the balloons (V₂), the temperature (T₂ = 25.0 °C = 298.15 K), and the pressure (P₂ = 2.0 atm).
First, we need to convert the radius of the balloons from centimeters to meters:
Radius = 15 cm = 0.15 m
Next, we can calculate the volume of one balloon using the formula for the volume of a sphere:
V = (4/3) * π * r³
V = (4/3) * 3.14159 * (0.15 m)³
V ≈ 0.14137 m³
Now, we can calculate the volume of gas in all 23 balloons:
Total Volume = 23 balloons * Volume per balloon
Total Volume = 23 * 0.14137 m³
Total Volume ≈ 3.2515 m³
Now that we have the final volume of gas in the balloons (V₂ = 3.2515 m³), we can use the combined gas law to find the initial volume of gas in the tank (V₁).
(P₁ * V₁) / T₁ = (P₂ * V₂) / T₂
(1 atm * 8.0 L) / 298.15 K = (2.0 atm * V₂) / 298.15 K
Cross-multiplying and rearranging the equation, we get:
V₁ = (2.0 atm * V₂) / 1 atm
V₁ = 2.0 * 3.2515 m³
V₁ ≈ 6.503 m³
Therefore, all the gas in the tank would occupy approximately 6.503 m³ at 2.0 atm and a temperature of 25.0 °C (when it is put into the balloons).