Question

One of the causes of the explosion at Chernobyl may have been the reaction between zirconium, which coated the fuel rods, and steam. $$ \mathrm{Zr}(s)+2 \mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{ZrO}_{2}(s)+2 \mathrm{H}_{2}(g) $$ If half a metric ton of zirconium reacted, what pressure was exerted by the hydrogen gas produced at \(55^{\circ} \mathrm{C}\) in the containment chamber, which had a volume of \(2.0 \times 10^{4} \mathrm{~L}\) ?

Short answer

Answer: The pressure exerted by the hydrogen gas is approximately 1.47 atm.

Step-by-step solution

Calculate the number of moles of zirconium reacted

First, we need to convert the mass of zirconium into moles. Using the molar mass of zirconium, which is 91.22 g/mol, we will perform this conversion: $$ \frac{0.5 \times 10^6 \, g}{91.22 \, g/mol} = 5478.62 \text{ moles of Zr} $$

Find the moles of hydrogen gas produced

According to the balanced reaction equation, 1 mole of Zr reacts with 2 moles of H2O to produce 2 moles of H2. Therefore, the moles of hydrogen gas produced will be twice the moles of zirconium that reacted: $$ 2 \times 5478.62 \, \text{moles of Zr} = 10957.24 \text{ moles of H}_2 $$

Convert the temperature to Kelvin

The Ideal Gas Law equation uses the temperature in Kelvin, so we need to convert the given temperature from Celsius to Kelvin: $$ T = 55^{\circ}C + 273.15 = 328.15 \, K $$

Apply the Ideal Gas Law equation

Now we have everything we need to find the pressure exerted by the hydrogen gas produced. Using the Ideal Gas Law equation \(PV = nRT\), and converting the given volume to liters ( \(2.0 \times 10^4 \, L\) ), we can solve for the pressure: $$ P = \frac{nRT}{V} = \frac{(10957.24 \, \text{moles}) \times (0.0821 \, L \, atm/(mol \, K)) \times (328.15 \, K)}{2.0 \times 10^{4} \, L} $$ $$ P ≈ 1.47 \, atm $$ Therefore, the pressure exerted by the hydrogen gas produced in the containment chamber is approximately 1.47 atm.

Key concepts

Understanding Chemical Reactions

Chemical reactions are fundamental processes in which reactants transform into products through the breaking and forming of chemical bonds. One historical example of a chemical reaction with dramatic consequences is the reaction between zirconium and steam at the Chernobyl nuclear plant. Here's how it happens:

• Zirconium, which was used to coat the fuel rods, can react with steam (water in the gas phase) under certain conditions.
• The reaction between zirconium (Zr) and water (H₂O) produces zirconium dioxide (ZrO₂) and hydrogen gas (H₂).
• From the reaction equation, you can tell the stoichiometry of the reaction – for each mole of Zr that reacts, two moles of H₂ gas are produced.

This type of reaction not only changes the chemical composition but also involves energy changes, which can sometimes be rapid and violent, contributing to disasters like Chernobyl.

Applying the Ideal Gas Law

The Ideal Gas Law is a crucial equation in chemistry that relates the pressure (P), volume (V), temperature (T), and the number of moles (n) of an ideal gas. It is represented as:\[ PV = nRT \]where R is the ideal gas constant. To apply this to a real-life situation like the one that occurred at Chernobyl, follow these crucial steps:

• Knowing the moles of gas involved and the volume of the containment chamber is essential.
• Temperature must be in Kelvin, which is Celsius plus 273.15. For Chernobyl, a temperature of \(55^{\text{o}}C \) converts to \( 328.15 K \)
• With these values, the pressure exerted by the gas can be calculated, critical information for understanding the impact of the chemical reaction in a closed environment, such as the containment chamber in a nuclear reactor.

Molar Mass Calculation

Molar mass is an intrinsic property of substances and is defined as the mass of a given substance (chemical element or chemical compound) divided by the amount of substance. It is expressed in units of grams per mole (g/mol).

To calculate the molar mass:

• Identify the atomic mass of each element in the molecule from the periodic table.
• Add the atomic masses according to each element's proportion in the compound.

In the Chernobyl disaster scenario, the molar mass of zirconium is 91.22 g/mol, which is used to convert the mass of zirconium into moles. Knowing the molar mass and the mass, you can find out the number of moles, an essential step for using the ideal gas law equation. Accurate molar mass calculation is critical for predicting the quantities of products and reactants in chemical reactions.