Question

A sample of zinc metal reacts completely with an excess of hydrochloric acid: $$ \mathrm{Zn}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{ZnCl}_{2}(a q)+\mathrm{H}_{2}(g) $$ The hydrogen gas produced is collected over water at \(25.0^{\circ} \mathrm{C}\) using an arrangement similar to that shown in Figure \(10.14(\mathrm{a})\). The volume of the gas is \(7.80 \mathrm{~L},\) and the pressure is 0.980 atm. Calculate the amount of zinc metal in grams consumed in the reaction. (Vapor pressure of water at \(\left.25^{\circ} \mathrm{C}=23.8 \mathrm{mmHg} .\right)\)

Short answer

19.8 grams of zinc are consumed.

Step-by-step solution

Convert Vapor Pressure of Water to Atmospheres

First, convert the vapor pressure of water from mmHg to atm. The vapor pressure of water at 25°C is given as 23.8 mmHg. To convert this value to atmospheres, use the conversion factor: 1 atm = 760 mmHg.So, the vapor pressure of water in atm is given by:\[ \text{Vapor Pressure in atm} = \frac{23.8 \text{ mmHg}}{760 \text{ mmHg/atm}} = 0.0313 \text{ atm} \]

Calculate Pressure of Dry Hydrogen Gas

Next, find the pressure of the dry hydrogen gas by subtracting the vapor pressure of water from the total pressure measured over water.\[ P_{\text{dry }} = P_{\text{total}} - P_{\text{water}} = 0.980 \text{ atm} - 0.0313 \text{ atm} = 0.9487 \text{ atm} \]

Use Ideal Gas Law to Calculate Moles of Hydrogen

Next, use the Ideal Gas Law to calculate the moles of hydrogen gas produced.The Ideal Gas Law is \( PV = nRT \), where:- \( P = 0.9487 \) atm (pressure of dry hydrogen gas),- \( V = 7.80 \) L (volume of gas),- \( R = 0.0821 \) L·atm/mol·K (ideal gas constant),- \( T = 25.0°C = 298 \) K (temperature in Kelvin; conversion is done by adding 273 to Celsius).Plug these values into the equation:\[ n = \frac{PV}{RT} = \frac{0.9487 \cdot 7.80}{0.0821 \cdot 298} \approx 0.303 \text{ moles of } \text{H}_2 \]

Determine Moles of Zinc Consumed

From the balanced chemical equation, we know that 1 mol of Zn produces 1 mol of \( \text{H}_2 \). Therefore, the moles of Zn consumed are equal to the moles of \( \text{H}_2 \) produced:\[ n_{\text{Zn}} = 0.303 \text{ moles} \]

Convert Moles of Zinc to Grams

Finally, convert the moles of zinc to grams using the molar mass of zinc, which is 65.38 g/mol.\[ \text{Mass of Zn} = n \cdot \text{molar mass of Zn} = 0.303 \cdot 65.38 \approx 19.8 \text{ grams} \]

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry that relates the pressure, volume, temperature, and amount of gas. It is expressed through the equation \( PV = nRT \). Each letter represents a different property of the gas:

• \( P \) is the pressure, measured in atmospheres (atm).
• \( V \) is the volume, typically in liters (L).
• \( n \) is the number of moles of gas.
• \( R \) is the ideal gas constant, valued at 0.0821 L·atm/mol·K.
• \( T \) is the temperature in Kelvin (K), which you obtain by adding 273 to the Celsius temperature.

In the given exercise, the ideal gas law helps us to find out the moles of hydrogen gas by knowing the pressure after adjusting for vapor pressure, volume of the gas collected, and the temperature. This is a crucial step that connects gas properties to real-world chemical reactions.

Stoichiometry

Stoichiometry is the part of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It is used to balance chemical equations and relate the amounts of substances that react or are produced.
In the current exercise, stoichiometry is applied once we have determined the amount of hydrogen gas using the ideal gas law. The balanced equation for the reaction is: \[ \mathrm{Zn}(s) + 2 \mathrm{HCl}(aq) \rightarrow \mathrm{ZnCl}_2(aq) + \mathrm{H}_2(g) \]
From this equation, one mole of zinc reacts with two moles of hydrochloric acid to produce one mole of hydrogen gas. By knowing how much hydrogen gas is formed, we directly deduce the number of moles of zinc consumed, as the ratio of zinc to hydrogen in the reaction is 1:1.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid form. In the context of gas collection over water, it refers to the pressure of water vapor present and must be accounted for when measuring the total pressure of a gas.
In the exercise, the hydrogen gas is collected over water at \( 25.0^{\circ} \mathrm{C} \), meaning the measured pressure includes both the gas pressure and the vapor pressure of water. The given vapor pressure of water at this temperature is \( 23.8 \mathrm{mmHg} \), which is converted to atmospheres (\( 0.0313 \mathrm{atm} \)) to be used in the ideal gas law calculations. Adjusting for this is essential for finding the true pressure of the collected hydrogen gas.

Molar Mass

Molar mass is the mass of a given substance (molecule or atom) divided by the amount of substance. For elements, it is typically expressed in grams per mole (g/mol), which tells you the weight of one mole of an element or compound.
In the reaction, zinc's molar mass is necessary for converting moles of zinc calculated from stoichiometry into grams, allowing us to know the actual amount of zinc consumed in the reaction. For zinc, the molar mass used is \( 65.38 \mathrm{g/mol} \). Thus, by multiplying the moles of zinc consumed by this molar mass, we obtain the mass in grams.

Chemical Equation Balancing

Balancing chemical equations is critical because it ensures that the same number of each type of atom is present on both sides of the equation, following the law of conservation of mass.
In this exercise, the chemical equation \[ \mathrm{Zn}(s) + 2 \mathrm{HCl}(aq) \rightarrow \mathrm{ZnCl}_2(aq) + \mathrm{H}_2(g) \] describes the reaction between zinc and hydrochloric acid.
Here, it is essential to balance it to accurately reflect the stoichiometric relationships:

• One zinc atom reacts with two hydrochloric acid molecules.
• This produces one zinc chloride molecule and one hydrogen gas molecule.

This balancing is a straightforward example, but sets the groundwork for more complex reactions where you might have several substances reacting in different proportions.