To determine the approximate temperature of the fireball after it expands adiabatically to a radius of 1000 meters, we can use the principles of thermodynamics, specifically the adiabatic process for an ideal gas. In an adiabatic process, there is no heat exchange with the surroundings, and we can apply the relationship between temperature and volume.
Understanding Adiabatic Expansion
In an adiabatic process, the temperature and volume of a gas are related by the equation:
T1 * V1^(γ-1) = T2 * V2^(γ-1)
Where:
- T1 = initial temperature
- V1 = initial volume
- T2 = final temperature
- V2 = final volume
- γ = heat capacity ratio (given as 1.66)
Calculating Initial and Final Volumes
First, we need to calculate the initial and final volumes of the fireball. The volume of a sphere is given by the formula:
V = (4/3) * π * r³
For the initial radius (r1 = 100 m):
- V1 = (4/3) * π * (100)³
- V1 ≈ 4188000 m³
For the final radius (r2 = 1000 m):
- V2 = (4/3) * π * (1000)³
- V2 ≈ 4188000000 m³
Applying the Adiabatic Relation
Now we can substitute the values into the adiabatic equation. We know:
- T1 = 1000 K
- V1 ≈ 4188000 m³
- V2 ≈ 4188000000 m³
- γ = 1.66
Substituting these values into the equation:
1000 * (4188000)^(1.66 - 1) = T2 * (4188000000)^(1.66 - 1)
This simplifies to:
1000 * (4188000)^(0.66) = T2 * (4188000000)^(0.66)
Calculating the Temperature
Now, we can calculate the left and right sides of the equation:
Calculating the volumes raised to the power of 0.66:
- (4188000)^(0.66) ≈ 1740000
- (4188000000)^(0.66) ≈ 1740000000
Now substituting these values back into the equation:
1000 * 1740000 = T2 * 1740000000
Solving for T2:
T2 = (1000 * 1740000) / 1740000000
T2 ≈ 1000 * 0.1 = 100 K
Final Result
Thus, after the adiabatic expansion of the fireball to a radius of 1000 meters, the approximate temperature will be around 100 K. This significant drop in temperature illustrates how adiabatic expansion leads to cooling, which is a fundamental concept in thermodynamics.
