A 1.50- gas sample at 722 and 25.0 contains 7.56 radon-220 by volume. Radon-220 is an alpha-emitter with a half-life of 55.6. How many alpha particles are emitted by the gas sample in 3.00 minutes?

All of the numbers need units.

To calculate the number of alpha particles emitted by the gas sample in 3.00 minutes, we will use the information provided about the gas sample and the properties of radon-220.

First, let's break down the given information:

- Gas sample volume: 1.50 L
- Temperature: 722 K
- Pressure: 25.0 atm
- Radon-220 concentration: 7.56 (measured in a volume basis)

To calculate the number of alpha particles emitted, we need to determine the number of radon-220 atoms present in the gas sample. From there, we can calculate the number of alpha particles based on the half-life and decay process.

Here are the steps to find the solution:

Step 1: Calculate the number of radon-220 atoms in the gas sample.
- Convert the gas sample volume from liters (L) to moles (mol) using the ideal gas law.
- Use Avogadro's number (6.022 x 10^23) to convert moles to the number of atoms.

Step 2: Calculate the decay constant.
- The half-life (t1/2) of radon-220 is given as 55.6 seconds.
- The decay constant (λ) can be calculated using the equation λ = ln(2) / t1/2.

Step 3: Calculate the number of alpha particles emitted in 3.00 minutes.
- Since the half-life provided is in seconds, convert 3.00 minutes to seconds.
- Use the decay equation N(t) = N(0) * e^(-λt) to calculate the remaining number of radon-220 atoms after 3.00 minutes.
- The number of alpha particles emitted can be found by subtracting the remaining number of radon-220 atoms from the initial number of radon-220 atoms and multiplying by 2 (since each radon-220 atom emits two alpha particles).

Now, let's follow these steps to find the solution:

Step 1: Calculating the number of radon-220 atoms in the gas sample:
- Convert the gas sample volume from liters (L) to moles (mol) using the ideal gas law:
PV = nRT
n = PV / RT
n = (25.0 atm) * (1.50 L) / [(0.0821 L.atm/mol.K) * (722 K)]
n ≈ 0.540 mol

- Use Avogadro's number (6.022 x 10^23) to convert moles to the number of atoms:
Number of radon-220 atoms = (0.540 mol) * (6.022 x 10^23 atoms/mol) ≈ 3.252 x 10^23 atoms

Step 2: Calculate the decay constant (λ):
- λ = ln(2) / t1/2
λ = ln(2) / 55.6 s
λ ≈ 0.0125 s^-1

Step 3: Calculate the number of alpha particles emitted in 3.00 minutes:
- Convert 3.00 minutes to seconds:
3.00 minutes * (60 s/1 minute) = 180 seconds

- Use the decay equation N(t) = N(0) * e^(-λt):
N(t) = (3.252 x 10^23 atoms) * e^(-0.0125 s^-1 * 180 s)

- Calculate the remaining number of radon-220 atoms:
N(t) ≈ 1.036 x 10^20 atoms

- Calculate the number of alpha particles emitted:
Number of alpha particles = (3.252 x 10^23 atoms - 1.036 x 10^20 atoms) * 2 ≈ 6.412 x 10^23 alpha particles

Therefore, the gas sample would emit approximately 6.412 x 10^23 alpha particles in 3.00 minutes.