Problem 71 - Chemistry

Question

A piece of sodium metal reacts completely with water as follows: $$ 2 \mathrm{Na}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow 2 \mathrm{NaOH}(a q)+\mathrm{H}_{2}(g) $$ The hydrogen gas generated is collected over water at \(25.0^{\circ} \mathrm{C}\). The volume of the gas is \(246 \mathrm{~mL}\) measured at 1.00 atm. Calculate the number of grams of sodium used in the reaction. (Vapor pressure of water at \(25^{\circ} \mathrm{C}=0.0313\) atm. \()\)

Step-by-Step Solution

Answer

0.2229 g of sodium was used in the reaction.

Step 1: Calculate pressure of hydrogen gas

The total pressure is the pressure of the hydrogen gas plus the pressure of the water vapor. Therefore, the pressure of the hydrogen gas is the total pressure minus the vapor pressure of the water, which is calculated as follows: \( P_{H2} = P_{total} - P_{water} = 1.00 atm - 0.0313 atm = 0.9687 atm \)

Step 2: Convert volume to litres

The volume of the hydrogen gas is given in milliliters, but for calculation involving the ideal gas law, it should be in liters. Therefore, convert 246 mL to liters: \( V_{H2} = 246 mL * (1 L / 1000 mL) = 0.246 L \)

Step 3: Calculate temperature in Kelvin

The temperature for calculation involving the ideal gas law should be in Kelvin, not Celsius. To convert from degrees Celsius to Kelvin, add 273.15: \( T_{H2} = 25.0^{\circ}C + 273.15 = 298.15 K \)

Step 4: Calculate the number of moles of H2

Use the ideal gas law \( PV=nRT \), where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature, to calculate the number of moles of hydrogen gas (n_{H2}): \( n_{H2} = \frac{P_{H2} * V_{H2}}{R * T_{H2}} = \frac{0.9687 atm * 0.246 L}{0.0821 L atm mol^{-1} K^{-1} * 298.15 K} = 0.0097 mol \)

Step 5: Calculate the number of moles of Na

From the balanced chemical equation, it can be seen that the number of moles of sodium reacted (n_{Na}) is the same as the number of moles of hydrogen gas produced. Therefore, \( n_{Na} = n_{H2} = 0.0097 mol \)

Step 6: Calculate mass of sodium

Using the molar mass of sodium (\( M_{Na} = 22.99 g/mol \)), the mass of sodium reacted can be calculated: \( mass_{Na} = n_{Na} * M_{Na} = 0.0097 mol * 22.99 g/mol = 0.2229 g \)

Key Concepts

Chemical Reactions

Understanding chemical reactions is critical in chemistry. A chemical reaction involves the transformation of one set of chemical substances to another. We represent these transformations through chemical equations, which like in the case of sodium reacting with water, demonstrate how reactants (sodium metal and water) convert to products (sodium hydroxide and hydrogen gas).

Properly balancing these equations is crucial, as it respects the law of conservation of mass, ensuring that the number of atoms for each element is the same on both sides of the equation.

Stoichiometry

Stoichiometry is the study of the quantitative relationships, or ratios, between substances as they participate in chemical reactions. It allows us to predict the amount of products that will form in a given reaction based on the quantity of reactants. For the sodium metal reaction, stoichiometry helps us understand the mole ratio between sodium and hydrogen gas, which is 1:1 according to the balanced chemical equation. This means that for every mole of sodium that reacts, one mole of hydrogen gas is produced.

The Ideal Gas Law

The ideal gas law is a fundamental equation that relates the pressure (P), volume (V), temperature (T), and amount (n) of moles of a gas through the formula PV = nRT. Here R is the ideal gas constant. This law assumes that gas molecules are point particles that move randomly with no intermolecular forces.

When solving problems involving gases, it's often necessary to work in the correct units, typically atmospheres for pressure, liters for volume, Kelvin for temperature, and moles for the amount of substance. Adjusting conditions to STP (standard temperature and pressure) or using the given conditions as in the sodium metal reaction enhances the accuracy of the calculations.

Mole Concept

The mole concept is a bridge between the macroscopic world we can measure and the microscopic world of atoms and molecules. One mole is Avogadro's number (approximately 6.022 x 10^23) of particles, which is analogous to a 'dozen' but for an incredibly large quantity.

Using the mole concept allows us to convert between mass, atoms, and molecular amounts. In the sodium reaction example, we calculated the moles of hydrogen gas generated and used this information to infer the moles, and ultimately the mass, of sodium that reacted.