A 15.7-L tank is filled with H2 to a pressure of 237 atm. How many balloons (each 2.00 L) can be inflated to a pressure of 1.00 atm from the tank? Assume that there is no temperature change and that the tank cannot be emptied below 1.00 atm pressure.
We are given:
- volume of the tank (V1) = 15.7 L
- initial pressure in the tank (P1) = 237 atm
- final pressure in the tank (P2) = 1.00 atm
- volume of each balloon (V_balloon) = 2.00 L
- pressure in each balloon (P_balloon) = 1.00 atm
According to Boyle's law, the product of the initial pressure and volume of the gas is equal to the product of the final pressure and volume of the gas when the temperature remains constant. Therefore, (P1 * V1) = (P2 * V2).
Rearranging to find the final volume (V2), we get:
V2 = (P1 * V1) / P2
Now we can plug in the given values:
V2 = (237 atm * 15.7 L) / 1.00 atm
V2 = 3720.9 L
Now, we need to find the difference in volume between the initial and final states of the tank, which would be the volume available for inflation:
V_available = V2 - V1
V_available = 3720.9 L - 15.7 L
V_available = 3705.2 L
Finally, we need to calculate how many balloons can be inflated with this available volume. Since each balloon has a volume of 2.00 L, we can divide the available volume by the volume of one balloon:
No. of balloons = V_available / V_balloon
No. of balloons = 3705.2 L / 2.00 L
No. of balloons = 1852.6
Since we can't have a fraction of a balloon, we round down to the nearest whole number:
No. of balloons = 1852
To determine how many balloons can be inflated using the given tank, we need to calculate the number of moles of hydrogen gas in the tank and then divide that by the number of moles required to inflate one balloon.
First, let's calculate the number of moles of hydrogen gas in the tank using the ideal gas law equation:
PV = nRT
where:
P = pressure (in atm)
V = volume (in liters)
n = number of moles
R = ideal gas constant (0.0821 L·atm/(mol·K))
T = temperature (in Kelvin)
Rearranging the equation to solve for n:
n = PV / RT
Substituting the given values:
P = 237 atm
V = 15.7 L
R = 0.0821 L·atm/(mol·K)
(assuming the temperature remains constant, we can neglect it for this calculation)
n = (237 atm) * (15.7 L) / (0.0821 L·atm/(mol·K))
Now, let's calculate the number of moles required to inflate one balloon. Since the volume of one balloon is given as 2.00 L, we can use the same equation:
n_b = PV / RT
Substituting the given values:
P = 1.00 atm
V = 2.00 L
R = 0.0821 L·atm/(mol·K)
n_b = (1.00 atm) * (2.00 L) / (0.0821 L·atm/(mol·K))
Finally, we can determine the number of balloons that can be inflated from the tank by dividing the number of moles of hydrogen gas in the tank by the number of moles required to inflate one balloon:
Number of balloons = n / n_b
To determine how many balloons can be inflated from the tank, we need to find the total volume of hydrogen gas that is available in the tank, and then divide it by the volume of each balloon.
First, let's convert the pressure of the hydrogen gas in the tank from atm to Pascals (Pa), which is the SI unit for pressure:
1 atm = 101325 Pa
237 atm * 101325 Pa/atm = 2.4 x 10^7 Pa
Next, we need to convert the volume of the tank from liters to cubic meters (m³), since we'll be using SI units for the calculations:
15.7 L = 0.0157 m³
We can then use the ideal gas law to find the number of moles of hydrogen gas in the tank. The ideal gas law equation is:
PV = nRT
Where:
P = pressure (Pa)
V = volume (m³)
n = number of moles
R = ideal gas constant (8.314 J/(mol·K))
T = temperature (Kelvin)
Since the temperature is not given and there is no temperature change, we can assume it is constant. So the equation becomes:
PV = nRT
Solving for n (number of moles):
n = PV / RT
n = (2.4 x 10^7 Pa) * (0.0157 m³) / (8.314 J/(mol·K) * T)
Since we are only interested in the number of balloons that can be inflated to a pressure of 1.00 atm, we will consider the final pressure to be 1.00 atm or 101325 Pa.
Now, let's calculate the number of moles of hydrogen gas:
n = (2.4 x 10^7 Pa) * (0.0157 m³) / (8.314 J/(mol·K) * T)
To simplify the calculation, let's assume the temperature is 298 K:
n = (2.4 x 10^7 Pa) * (0.0157 m³) / (8.314 J/(mol·K) * 298 K)
n ≈ 119.32 moles
Finally, we can calculate the total volume of the hydrogen gas in the tank:
V_total = n * R * T / P
V_total = (119.32 moles) * (8.314 J/(mol·K)) * (298 K) / (2.4 x 10^7 Pa)
V_total ≈ 1.57 m³
Now, we can find the number of balloons that can be inflated to a pressure of 1.00 atm (0.101325 Pa) from the tank:
Number of balloons = V_total / V_balloon
Number of balloons = 1.57 m³ / 2.00 L
However, we need to convert liters to cubic meters:
Number of balloons = 1.57 m³ / (2.00 x 10^-3 m³)
Number of balloons ≈ 785 balloons
Therefore, approximately 785 balloons (each with a volume of 2.00 L) can be inflated to a pressure of 1.00 atm from the tank.