Question

The reaction of zinc metal with hydrochloric acid was used to produce \(1.5\) moles of hydrogen gas at \(298 \mathrm{~K}\) and 1 atm pressure. The magnitude work done in pushing back the atmosphere is (a) \(596 \mathrm{cal}\) (b) \(894 \mathrm{cal}\) (c) \(447 \mathrm{cal}\) (d) \(298 \mathrm{cal}\)

Short answer

447 \text{cal}

Step-by-step solution

Identify the type of work done

The work done by the gas against atmospheric pressure is a type of mechanical work that can be calculated by the formula: work done, w = -PΔV, where P is the pressure of the gas and ΔV is the change in volume.

Calculate the change in volume, ΔV

Using the ideal gas law, PV = nRT, where n is the number of moles of the gas, R is the ideal gas constant (0.0821 atm·L/mol·K), and T is the temperature in Kelvin, we can calculate the change in volume produced by 1.5 moles of hydrogen gas. ΔV = (nRT) / P. Substitute n = 1.5 moles, R = 0.0821 atm·L/mol·K, T = 298 K, and P = 1 atm.

Calculate the work done

Using the calculated ΔV from Step 2, plug the values into the equation w = -PΔV to get the work done in liters·atm. Convert the calculated work from liters·atm to calories using the conversion factor 1 liter·atm = 24.2 cal.

Choose the correct answer

Compare the calculated value of work in calories with the given options to identify the correct answer.

Key concepts

Understanding Chemical Work Calculation

Chemical work involves the energy transfer that occurs when a system exerts a force through a distance, often seen in reactions involving gases where an expansion or compression of gas takes place. In physical chemistry, it's important to be able to calculate this work, which is usually carried out by or against the pressure of the surroundings.

A well-known formula for this calculation is the negative pressure-volume work formula: \[\begin{equation}w = -P\triangle V\text{,}\thinspace\end{equation}\]where \(w\) represents work done, \(P\) is the constant pressure, and \(\triangle V\) denotes the change in volume of the gas. The negative sign indicates the work done by the system (gas) when it expands, as it must work against atmospheric pressure.

The distinction between work done by the system, which is counted as negative, and work done on the system, which is positive, is a crucial concept. When a gas expands and does work on its surroundings, the internal energy of the gas decreases, and so the work is considered negative in the context of the first law of thermodynamics.

Applying the Ideal Gas Law

To calculate the chemical work, particularly during a gas expansion or compression, the Ideal Gas Law provides a bridge between the macroscopic and microscopic properties of the gas. The Ideal Gas Law is represented by the equation:\[\begin{equation}PV = nRT\text{,}\thinspace\end{equation}\]where \(P\) stands for pressure, \(V\) is volume, \(n\) is the number of moles of the gas, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin. By manipulating this equation, we can derive the change in volume (\(\triangle V\)) under constant pressure and temperature conditions, which is essential for our work calculation.

In the given exercise, tying the Ideal Gas Law to the work calculation is fundamental. By inserting the known values for the number of moles of hydrogen gas produced (1.5 moles), temperature (298 K), the gas constant (0.0821 atm·L/mol·K), and the pressure (1 atm), students can solve for \(\triangle V\) which is crucial for finding the work done during the gas's expansion.

Navigating Physical Chemistry Exercises

Physical chemistry combines the principles of physics and chemistry to explain how chemical systems behave. To navigate exercises within this discipline effectively, a clear understanding of thermodynamics, kinetics, and quantum mechanics is required.

Exercises like the aforementioned chemical work calculation serve to solidify comprehension of these topics. It's about applying formulas accurately and understanding the physical significance of each variable in the equation. The ability to connect theoretical knowledge with practical examples is integral to mastering physical chemistry.

Stepwise problem-solving is a great technique for students—breaking down complex problems into manageable steps helps reduce errors and facilitates deeper understanding. When faced with such exercises, always begin by identifying known values and the required outcome, followed by systematically applying the relevant principles and formulas. Regular practice with a variety of problems will build the confidence and proficiency necessary to tackle the rigors of physical chemistry.