Question

How many liters of hydrogen gas are collected over water at \(18^{\circ} \mathrm{C}\) and \(725 \mathrm{mmHg}\) when \(0.84 \mathrm{~g}\) of lithium reacts with water? Aqueous lithium hydroxide also forms.

Short answer

1.74 liters of hydrogen gas

Step-by-step solution

- Write the balanced chemical equation

First, write the balanced chemical equation for the reaction of lithium with water:Li + H₂O → LiOH + H₂Thus, the balanced equation is: 2Li + 2H₂O → 2LiOH + H₂

- Calculate moles of lithium

Use the given mass of lithium and its molar mass to calculate the number of moles of lithium reacting:Molar mass of lithium (Li) = 6.94 g/molMoles of Li = \(\frac{0.84 \, \text{g}}{6.94 \, \text{g/mol}}\) = 0.121 moles

- Calculate moles of hydrogen gas produced

From the balanced equation, 2 moles of Li produce 1 mole of H₂ gas. Calculate the moles of H₂ gas produced:Moles of H₂ = \(\frac{0.121 \text{ moles Li} \times 1 \text{ mole H₂}}{2 \text{ moles Li}}\) = 0.0605 moles H₂

- Convert moles of H₂ to volume

Use the ideal gas law equation to find the volume of hydrogen gas at the given conditions. The ideal gas law is PV = nRTwhere P is the pressure (converted to atm), V is the volume in liters, n is the number of moles, R is the gas constant (0.0821 L·atm/(mol·K)), and T is the temperature in Kelvin.

- Adjust for water vapor pressure

Since the gas is collected over water, adjust the pressure to account for water vapor pressure. At 18°C, the vapor pressure of water is 15.5 mmHg. The corrected pressure is:P_total = 725 mmHg P_H2 = P_total - P_vapor = 725 mmHg - 15.5 mmHg = 709.5 mmHg Convert this to atm: P_H2 in atm = \(\frac{709.5}{760}\)

- Calculate the volume of H₂ gas

Convert the temperature to Kelvin: T = 18°C + 273 = 291 K Using the ideal gas law to find the volume:\[ V = \frac{nRT}{P} \]\[ V = \frac{0.0605 \text{ moles} \times 0.0821 \text{ L·atm/mol·K} \times 291 \text{ K}}{\frac{709.5}{760} \text{ atm}} \]V ≈ 1.74 L

Key concepts

Stoichiometry

Stoichiometry helps us understand the quantitative relationships in a chemical reaction. It is crucial for predicting the amount of products formed from given reactants. In the problem of lithium reacting with water, we start with a known mass of lithium. By converting this mass into moles using lithium's molar mass, we can use the coefficients from the balanced chemical equation to find the moles of hydrogen gas produced. In simpler words, stoichiometry ensures that we know 'how much' reactants we need and 'how much' product we will get.

Balanced Chemical Equation

A balanced chemical equation shows the exact proportions of reactants and products in a chemical reaction. It's essential because it follows the law of conservation of mass, meaning no atoms are lost or gained during the reaction. For instance, in our exercise, the balanced chemical equation for the reaction of lithium with water is: 2Li + 2H₂O → 2LiOH + H₂. This equation tells us that 2 moles of lithium react with 2 moles of water to produce 2 moles of lithium hydroxide and 1 mole of hydrogen gas. Balancing chemical equations makes sure that we have a clear, quantifiable map of the reaction.

Molar Volume of Gas

The molar volume of gas concept is integral to understanding gas behavior at standard temperature and pressure (STP). It tells us that 1 mole of any ideal gas occupies 22.4 liters at STP. In our exercise, we are interested in finding out how many liters of hydrogen gas are produced. To get this, we first find the moles of hydrogen gas from the stoichiometry calculations. Then, using the ideal gas law (PV = nRT), we can determine the volume of hydrogen gas under the given conditions by substituting the values we have.

Water Vapor Pressure

When collecting gas over water, it's important to account for water vapor because it contributes to the total pressure. Water vapor pressure varies with temperature. At 18°C, water vapor pressure is 15.5 mmHg. We subtract this from the total pressure to get the pressure exerted only by the collected gas. In our exercise, we adjusted the given total pressure of 725 mmHg by subtracting 15.5 mmHg, resulting in 709.5 mmHg for the hydrogen gas. This adjustment ensures we accurately use the ideal gas law to find the gas volume.