40 years ago, the first time a vehicle was driven over an extraterrestrial surface. The Soviet Lunokhod (moon walker) rover covered several miles on the moon, heated at night by a polonium-210 source, which undergoes an alpha decay to lead (Pb) yielding 5058 keV. Assume a power requirement of 100W.
a) Write the chemical equation for this nuclear decay.
b) How many Becquerel do you need in the polonium sample to achieve the required heating?
c) How long does this decay need to proceed at the same rate, so that 1 ug is converted into energy (c=3*10^8 m/s)? In reality, the rover 'died' from the activity in its polonium cell causing it to freeze out at night.
a) The chemical equation for the nuclear decay of polonium-210 is:
Po-210 -> Pb-206 + alpha particle
b) To determine the number of Becquerels (Bq) required in the polonium sample to achieve the required heating, we need to consider the power requirement and the energy released per decay.
Each alpha decay of polonium-210 releases energy of 5058 keV, which is equivalent to 8.084 × 10^-13 joules (since 1 eV = 1.602 x 10^-19 joules).
Power can be calculated as energy divided by time. In this case, the power requirement is given as 100 W. Thus, we have:
100 W = energy / time
We can rearrange this equation to solve for the time taken per decay:
time = energy / power
Since one decay produces 8.084 × 10^-13 joules of energy, we can substitute this value into the equation:
time = (8.084 × 10^-13 joules) / (100 W)
Using the definition of the Becquerel (Bq) as one decay per second, we can convert the time to Bq:
number of Bq = 1 / time
Substituting the calculated value for time, we can find the number of Bq required in the polonium sample.
c) To calculate the time necessary for the decay to proceed at a constant rate so that 1 ug (microgram) of polonium is converted into energy, we follow a similar approach.
First, we need to determine the number of polonium-210 atoms in 1 ug. The molar mass of Po-210 is approximately 210, so 1 mole of Po-210 is equal to 210 grams. We can then calculate the number of moles in 1 ug:
number of moles = (mass of sample) / (molar mass)
= (1 ug) / (210 g/mol)
= (1 × 10^-9 g) / (210 g/mol)
= 4.76 × 10^-12 moles
Next, we need to determine the number of atoms in 1 mole. Avogadro's number (Na) states that there are approximately 6.022 × 10^23 atoms per mole. So, the number of polonium-210 atoms in 1 ug is:
number of atoms = (number of moles) × (Na)
= (4.76 × 10^-12 moles) × (6.022 × 10^23 atoms/mol)
= 2.87 × 10^12 atoms
Since each decay of polonium-210 releases 8.084 × 10^-13 joules of energy, the total energy released from these atoms is:
total energy = (number of atoms) × (energy released per decay)
= (2.87 × 10^12 atoms) × (8.084 × 10^-13 joules/atom)
To find the time required for this decay to proceed at the same rate:
time = (total energy) / (power)
= [(2.87 × 10^12 atoms) × (8.084 × 10^-13 joules/atom)] / (100 W)
Using the value of the speed of light (c = 3 × 10^8 m/s), we can convert this time into seconds by dividing by the speed of light:
time in seconds = ([(2.87 × 10^12 atoms) × (8.084 × 10^-13 joules/atom)] / (100 W)) / (3 × 10^8 m/s)
This calculation will give us the time required for 1 ug of polonium-210 to be converted into energy at a constant rate.