Question

How many grams of sodium metal react with water to give \(75.0 \mathrm{~mL}\) of hydrogen gas at STP? $$\mathrm{Na}(\mathrm{s})+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{NaOH}(a +\mathrm{H}_{2}(g)$$

Short answer

0.1541 grams of sodium are required.

Step-by-step solution

Understand STP Conditions

Standard Temperature and Pressure (STP) conditions are defined as a temperature of 0°C (273.15 K) and a pressure of 1 atm. Under these conditions, 1 mole of any ideal gas occupies 22.4 liters.

Convert Volume of Hydrogen Gas to Moles

We need to determine the moles of hydrogen gas produced. The volume of hydrogen gas at STP is 75.0 mL. First, convert this volume to liters: \(75.0 \text{ mL} = 0.075 \text{ L}\). At STP, 1 mole of gas occupies 22.4 L, so the moles of hydrogen gas is \(\frac{0.075}{22.4} \approx 0.00335 \text{ moles of } H_2\).

Use Stoichiometry to Find Moles of Sodium

According to the balanced chemical equation, \(\mathrm{Na} \) and \( \mathrm{H}_2 \) are in a 2:1 mole ratio. Therefore, for every 1 mole of \( \mathrm{H}_2 \) produced, 2 moles of \( \mathrm{Na} \) are consumed. Hence, the moles of \( \mathrm{Na} \) required is \(2 \times 0.00335 = 0.00670\text{ moles of } \mathrm{Na}\).

Convert Moles of Sodium to Grams

The molar mass of sodium (\(\mathrm{Na}\)) is approximately 23.0 g/mol. Multiply the moles of sodium by its molar mass to find the mass in grams: \(0.00670 \times 23.0 = 0.1541 \text{ grams of sodium}\).

Conclusion

Thus, 0.1541 grams of sodium react with water to produce 75.0 mL of hydrogen gas at STP.

Key concepts

Understanding Chemical Reactions

Chemical reactions involve the transformation of one or more substances into different substances. In the case of the reaction between sodium metal and water, the equation you've seen describes the transformation where sodium (\(\mathrm{Na}\)) reacts with water (\(\mathrm{H}_2\mathrm{O}\)) to produce sodium hydroxide (\(\mathrm{NaOH}\)) and hydrogen gas (\(\mathrm{H}_2\)). This is a classic example of a single displacement reaction. It's important to ensure that chemical equations are balanced, meaning the same number of each type of atom is present on both sides of the reaction. This balance represents the law of conservation of mass, which states that mass cannot be created or destroyed in a chemical reaction. The rate and extent to which this reaction occurs can be influenced by external factors such as temperature and pressure, but this equation provides a clear foundational understanding of the chemical transformation.

• Reactants: Sodium (\(\mathrm{Na}\)) and Water (\(\mathrm{H}_2\mathrm{O}\))
• Products: Sodium Hydroxide (\(\mathrm{NaOH}\)) and Hydrogen Gas (\(\mathrm{H}_2\))

Exploring Molar Mass

Molar mass is the mass of a given substance (chemical element or chemical compound) divided by its amount of substance. It gives us the mass of one mole of a substance, expressed in grams per mole (g/mol). In chemistry, this is an essential concept because it connects the mass of a substance to the amount of that substance in moles, which is directly related to the number of molecules or atoms we have. For sodium (\(\mathrm{Na}\)), the molar mass is about 23.0 g/mol. This means that one mole of sodium atoms weighs 23 grams. To convert moles to grams, you multiply the number of moles by the molar mass. In chemical reactions, knowing the molar mass helps in determining how much of each reactant is needed and how much product can be expected to form.

• Formula: Mass (g) = Moles × Molar Mass (g/mol)
• Example: To convert 0.00670 moles of sodium to grams, calculate: \(0.00670 \times 23.0 = 0.1541\) grams.

Basics of Volume Conversion

Volume conversion is crucial when dealing with gases in chemical reactions. Different units like milliliters (mL) and liters (L) are commonly used. The key reason for converting mL to L is because the standard conditions for calculations, such as the ideal gas law, are in liters. To ensure correct mathematical treatment, speakers of equations or chemistry problems often must convert volumes. To convert milliliters to liters, you divide by 1000, since there are 1000 mL in one liter. In our specific problem, the volume of hydrogen gas given was 75.0 mL, which converts to 0.075 L. This step is necessary to proceed with calculations involving the ideal gas law or to find the moles of gas using the molar volume at STP. It simplifies the process and ensures that the right amount of each substance is used in a reaction.

• 1 L = 1000 mL
• Conversion Example: \(75.0\, mL \div 1000 = 0.075\, L\)

Understanding the Ideal Gas Law

The ideal gas law is fundamental in calculating and understanding gas behaviors under different conditions. It relates pressure, volume, temperature, and moles of a gas with the formula: \[ PV = nRT \]where:

• \(P\) is the pressure in atmospheres (atm)
• \(V\) is the volume in liters (L)
• \(n\) is the number of moles
• \(R\) is the ideal gas constant, which is approximately 0.0821 L·atm/mol·K
• \(T\) is the temperature in Kelvin (K)

The problem mentions standard temperature and pressure (STP) conditions, where 1 mole of gas occupies 22.4 liters, corresponding to 0°C and 1 atm. The ideal gas law helps predict how gases will react under these and other conditions. In our exercise, we used STP to find the moles of hydrogen gas, using the known volume of gas produced and the fact that at STP, 1 mole occupies 22.4 L. By using these conditions and the ideal gas relationship, it simplifies determining the quantities involved in reactions with gases.