High-pressure helium is available from gas producers in 0.045-m^3 cylinders at 400 bar and 298 K. Calculate the explosion equivalent of a tank of compressed helium in terms of kilograms of TNT. Assume helium is an ideal gas
To calculate the explosion equivalent, we need to determine the energy stored in the compressed helium gas. This can be done by finding the internal energy of the gas using the ideal gas law and then converting it to the explosive energy equivalent using a known value for TNT.
First, let's find the number of moles of helium gas present in the cylinder. We can use the ideal gas law equation:
PV = nRT
Where:
P = Pressure (in Pa)
V = Volume (in m^3)
n = Number of moles
R = Gas constant (8.314 J/mol·K)
T = Temperature (in K)
Converting the given values:
P = 400 bar = 400 x 10^5 Pa
V = 0.045 m^3
R = 8.314 J/mol·K
T = 298 K
Rearranging the equation to solve for n:
n = PV / RT
n = (400 x 10^5 Pa) * (0.045m^3) / (8.314 J/mol·K * 298 K)
n ≈ 2.398 mol
Next, let's calculate the internal energy (U) of the helium gas. The internal energy of an ideal gas is given by:
U = (3/2) nRT
U = (3/2) * (2.398 mol) * (8.314 J/mol·K) * (298 K)
U ≈ 8.776 kJ
Finally, we need to convert the internal energy of the helium gas to the explosive energy equivalent in terms of kilograms of TNT. The explosive energy of TNT is known to be approximately 4.184 megajoules per kilogram (MJ/kg).
Explosion Equivalent = U / (4.184 MJ/kg)
Explosion Equivalent = (8.776 kJ) / (4.184 MJ/kg)
Explosion Equivalent ≈ 0.0021 kg of TNT
So, approximately 0.0021 kilograms of TNT is the explosion equivalent of the compressed helium tank.
To calculate the explosion equivalent of a tank of compressed helium, we need to determine the amount of energy released by the expansion of the gas.
First, let's calculate the number of moles of helium in the cylinder. We can use the ideal gas law equation:
PV = nRT
Where:
P = pressure of the gas (400 bar)
V = volume of the gas (0.045 m^3)
n = number of moles of gas (unknown)
R = ideal gas constant (8.314 J/(mol·K))
T = temperature of the gas (298 K)
Rearranging the equation to solve for n, we get:
n = PV / RT
n = (400 * 10^5 Pa * 0.045 m^3) / (8.314 J/(mol·K) * 298 K)
n ≈ 0.0020249 mol
Next, we need to calculate the energy released by the expansion of the gas. The energy released is equal to the change in internal energy of the gas, which can be calculated using the equation:
ΔU = n * Cv * ΔT
Where:
ΔU = change in internal energy of the gas (J)
n = number of moles of gas (0.0020249 mol)
Cv = molar heat capacity at constant volume for helium (12.47 J/(mol·K))
ΔT = change in temperature (estimated to be 298 K)
ΔU = 0.0020249 mol * 12.47 J/(mol·K) * 298 K
ΔU ≈ 7.551 J
To calculate the explosion equivalent in terms of kilograms of TNT, we can use the conversion factor that 1 kilogram of TNT is equivalent to 4.18 × 10^9 J of energy.
Explosion Equivalent = ΔU / (4.18 × 10^9 J/kilogram)
Explosion Equivalent ≈ 7.551 J / (4.18 × 10^9 J/kg)
Explosion Equivalent ≈ 1.810 × 10^-12 kilograms of TNT
Therefore, the explosion equivalent of a tank of compressed helium is approximately 1.810 × 10^-12 kilograms of TNT.