Polonium-210, a naturally occurring radioisotope, is an alpha emitter, with t1/2=138 d. Assume that a sample fo 210Po with a mass of 0.700 mg was placed ina 250.0-mL flask, which was evacuated, sealed, and allowed to sit undisturbed. What would the pressure be inside the flask (in mmHg) at 20 degrees Celsius after 365 days if all the alpha particles emitted has become helium atoms?

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Step 1: Calculate the number of moles of 210Po initially present using the formula: \( n = \frac{mass}{molar\ mass} \). The molar mass of 210Po is approximately 210 g/mol.

Step 2: Determine the number of moles of 210Po remaining after 365 days using the decay formula: \( n_t = n_0 \times e^{-\lambda t} \), where \( \lambda = \frac{\ln(2)}{t_{1/2}} \) and \( t_{1/2} \) is the half-life of 210Po.

Step 3: Calculate the number of moles of 210Po that have decayed by subtracting the moles remaining from the initial moles: \( n_{decayed} = n_0 - n_t \).

Step 4: Since each decay of 210Po produces one helium atom, the moles of helium produced will be equal to the moles of 210Po that have decayed.

Step 5: Use the ideal gas law to calculate the pressure of the helium gas formed. The formula is \( P = \frac{nRT}{V} \), where \( n \) is the number of moles of helium, \( R \) is the gas constant (0.0821 L atm K^{-1} mol^{-1}), \( T \) is the temperature in Kelvin, and \( V \) is the volume of the flask in liters.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. In the case of Polonium-210, it undergoes alpha decay, releasing alpha particles that can transform into helium atoms. Understanding the half-life, which is the time required for half of the radioactive substance to decay, is crucial for calculating the remaining amount of Polonium-210 after a given period.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. In this scenario, after the decay of Polonium-210 into helium, the produced helium gas will exert pressure in the sealed flask. Knowing the volume of the flask and the temperature allows us to calculate the pressure exerted by the helium using the number of moles produced from the decay.

Molar Mass and Stoichiometry

Molar mass is the mass of one mole of a substance, which is essential for converting between grams and moles. In this problem, we need to determine how many moles of helium are produced from the decay of the initial mass of Polonium-210. Stoichiometry allows us to relate the amount of Polonium-210 that decays to the amount of helium produced, which is necessary for calculating the final pressure in the flask.

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