Question

In a constant-pressure calorimetry experiment, a \(2.675-\mathrm{g}\) piece of zinc metal is dropped into \(100.0 \mathrm{~mL}\) of \(1.75 \mathrm{M}\) hydrochloric acid in a closed vessel with a movable piston. The pressure and temperature in the laboratory are 769 torr and \(23.8^{\circ}\), respectively. Calculate the work done by the system.

Short answer

The work done by the system is approximately \(-101.0\) Joules.

Step-by-step solution

Convert Units

Convert the given pressure from torr to atm. Since 1 atm = 760 torr, we have:\[\text{Pressure in atm} = \frac{769}{760} = 1.0118 \text{ atm}\]

Calculate Moles of HCl

Calculate the moles of hydrochloric acid (HCl) present. Given the volume of HCl solution is 100.0 mL or 0.1 L and its molarity is 1.75 M, we use the formula:\[\text{Moles of HCl} = 1.75 \times 0.1 = 0.175 \text{ moles}\]

Write the Reaction Equation

The reaction between zinc and hydrochloric acid is:\[\text{Zn} + 2\text{HCl} \rightarrow \text{ZnCl}_2 + \text{H}_2\]Hence, 1 mole of Zn reacts with 2 moles of HCl.

Determine Limiting Reagent

Calculate the moles of Zn. The molar mass of Zn is approximately 65.38 g/mol, so:\[\text{Moles of Zn} = \frac{2.675}{65.38} \approx 0.0409 \text{ moles}\]Since 0.0409 moles of Zn require 0.0818 moles of HCl and there are 0.175 moles of HCl available, Zn is the limiting reactant.

Calculate Work Done

The work done by the system is related to the change in gas volume during the reaction. Given the stoichiometry of the reaction, the number of moles of gas produced can be expressed as:\[\text{Moles of } \text{H}_2 = 0.0409 \text{ (since 1 mole of Zn produces 1 mole of } \text{H}_2)\]Using the ideal gas law, and knowing that work done, \( W = -P\Delta V \), and for gases \( \Delta V = \frac{nRT}{P} \):\[W = -1.0118 \times 0.0409 \times 0.0821 \times (273.15 + 23.8) \]Calculate to find the work done:\[W \approx -1.0118 \times 0.0409 \times 0.0821 \times 296.95 \approx -0.996 \text{ L atm}\]Convert \(-0.996 \text{ L atm}\) to Joules, knowing that 1 L atm = 101.3 J:\[W \approx -101.0 \text{ J}\]

Conclusion

The work done by the system is approximately \(-101.0\) Joules.

Key concepts

Constant-Pressure Calorimetry

Constant-pressure calorimetry is a technique used to measure changes in enthalpy during chemical reactions conducted at atmospheric pressure. It allows us to determine how much heat is involved in a reaction. This method uses a device known as a calorimeter, which can measure temperature changes.

• The experiment is usually done in a closed vessel such as a cup or apparatus with a piston that can move as gases are produced or consumed.
• Since the pressure remains consistent, it simplifies the calculations needed to find the thermal energy absorbed or released.

Reactions that occur at constant pressure are common in laboratory experiments because they simulate open-air conditions. The goal is to track the change in temperature and relate this to the heat change in the system. This process is vital for understanding reaction energetics and designing processes in chemical engineering.

Ideal Gas Law

The Ideal Gas Law is an essential equation in chemistry that relates the pressure, volume, temperature, and amount of gas in moles. The formula is expressed as \( PV = nRT \), where:

• \(P\) stands for pressure in atm, \(V\) represents volume in liters, \(n\) is the number of moles, \(R\) is the ideal gas constant (0.0821 L atm/mol K), and \(T\) is temperature in Kelvin.

This fundamental equation allows us to calculate any one of these variables if the others are known.

It’s applicable when we assume gases are "ideal," meaning they follow the assumptions of no intermolecular forces and occupy no volume. While real gases deviate from ideality under certain conditions, this law provides a precise enough estimate for many practical purposes. It’s extensively used in laboratories to predict the behavior of gas during reactions, such as the generation of hydrogen gas in the reaction involving zinc and hydrochloric acid.

Limiting Reactant

In chemical reactions, the limiting reactant is the substance that is entirely consumed and limits the extent of the reaction. It dictates how much product can be formed. To determine which reactant is limiting, you compare the mole ratio of reactants used in the balanced chemical equation with the actual moles available.

• For example, in the reaction between zinc and hydrochloric acid, the formula \(\text{Zn} + 2\text{HCl} \rightarrow \text{ZnCl}_2 + \text{H}_2\) shows that one mole of Zn reacts with two moles of HCl.
• If there's less Zn relative to HCl according to the stoichiometric ratio, then Zn is the limiting reactant, as was found when only 0.0409 moles of Zn were available.

Identifying the limiting reactant is crucial because it influences both the yield of the product and calculates related properties such as work or heat change in reactions involving gaseous products.

Work Done by System

In thermodynamics, the work done by a system involving gases is associated with volume changes against a constant external pressure. The formula used is \( W = -P\Delta V \), where \( P \) is the pressure and \( \Delta V \) is the change in volume.

• Negative work indicates work done by the system, as seen in our example with the zinc and hydrochloric acid reaction. When hydrogen gas is produced, it expands the volume inside the chamber.
• This calculation employs the ideal gas law to find \( \Delta V \), involving temperature and moles of gas formed, then converts the answer into useful energy units like Joules (1 L atm equals 101.3 J).

Understanding the work done is essential in predicting how energy changes affect reaction feasibility and in designing industrial processes where energy efficiency is crucial.