Question

As 1 g of the radioactive element radium decays over 1 year, it produces 1.16 × 10¹⁸ alpha particles (helium nuclei). Each alpha particle becomes an atom of helium gas. What is the pressure in pascal of the helium gas produced if it occupies a volume of 125 mL at a temperature of 25 °C?

a) 2.78 kPa
b) 5.56 kPa
c) 11.12 kPa
d) 22.24 kPa

Answer 1

In conclusion, the pressure of the helium gas produced from the decay of 1 gram of the radioactive element radium, generating 1.16 × 10¹⁸ alpha particles (helium nuclei), is approximately 2.78 kPa. This pressure is calculated using the ideal gas law by converting the volume from milliliters to liters, determining the number of moles of helium from the given number of alpha particles, and adjusting for the temperature in kelvin. Therefore, the correct option is:

a) 2.78 kPa

Step-by-step explanation:

To find the pressure of the helium gas produced, we can use the ideal gas law, which is given by the equation:

`\[ PV = nRT \]`

where:

P is the pressure of the gas,

V is the volume of the gas,

n is the number of moles of gas,

R is the ideal gas constant, and

T is the temperature of the gas in kelvin.

First, let's convert the volume from milliliters to liters:

`\[ V = 125 \, \text{mL} * \frac{1 \, \text{L}}{1000 \, \text{mL}} = 0.125 \, \text{L} \]`

The number of moles n of helium gas can be determined from the given number of alpha particles produced by the decay of 1 g of radium. Since each alpha particle becomes an atom of helium gas, the number of moles of helium is equal to the number of alpha particles:

`\[ n = 1.16 * 10^(18) \, \text{mol} \]`

Now, we can use the ideal gas law to find the pressure:

`\[ P = (nRT)/(V) \]`

First, convert the temperature from Celsius to kelvin:

`\[ T = 25^0C} + 273.15 \,K^0C} = 298.15 ^0K} \]`

Now, substitute the values into the equation:

`\[ P = \frac{(1.16 * 10^(18) \, \text{mol}) * (8.314 \, \text{J/(mol K)}) * (298.15 \, \text{K})}{0.125 \, \text{L}} \]`

Calculate P:

`\[ P \approx 2.78 \, \text{kPa} \]`

So, the correct answer is:

a) 2.78 kPa