Question

A pellet of Zn of mass \(31.2 \mathrm{g}\) is dropped into a flask containing dilute \(\mathrm{H}_{2} \mathrm{SO}_{4}\) at a pressure of \(P=1.00\) bar and a temperature of \(T=300 .\) K. What is the reaction that occurs? Calculate \(w\) for the process.

Short answer

The reaction between zinc (Zn) and sulfuric acid (H₂SO₄) is given by the balanced chemical equation: \(Zn(s) + H_{2}SO_{4}(aq) \rightarrow ZnSO_{4}(aq) + H_{2}(g)\). Given the mass of Zn as 31.2 g, the number of moles is found to be 0.477 moles of both Zn and H₂ gas. The energy of the system with pressure 100,000 Pa and temperature 300 K is calculated as 1193.42 J using Ideal Gas Law. The work (w) for the process, determined by calculating the change in volume, is 0 J.

Step-by-step solution

Write the Balanced Chemical Equation

The reaction that occurs when Zn (zinc) comes in contact with H₂SO₄ (sulfuric acid) is: \[Zn(s) + H_{2}SO_{4}(aq) \rightarrow ZnSO_{4}(aq) + H_{2}(g)\] Now, let's calculate the number of moles of Zn and the energy for the given pressure and temperature.

Determine Moles and Energy

Given that the mass of Zn is 31.2 g. To calculate the number of moles (n), use the molar mass of Zn (65.38 g/mol): \[n = \frac{mass}{molar \thinspace mass} = \frac{31.2 \thinspace g}{65.38 \thinspace g/mol} = 0.477 \thinspace moles\] Using the balanced equation, we obtain 0.477 moles of H₂ gas. Pressure (P) = 1.00 bar = 100,000 Pa and Temperature (T) = 300 K. We need to convert the pressure and temperature into energy using the gas constant (R = 8.314 J/mol·K): \[\Delta U = nRT = (0.477 \thinspace moles)(8.314 J/mol \cdot K)(300 \thinspace K) = 1193.42 \thinspace J\]

Apply Ideal Gas Law to Calculate Work

Next, we will use the ideal gas law (PV=nRT) to calculate the change in volume and work done by the system. First, we need to calculate the initial and final volume of H₂ gas in the flask. Initial Volume (V₁): \[V_{1} = \frac{nRT}{P} = \frac{(0.477 \thinspace moles)(8.314 J/mol \cdot K)(300 \thinspace K)}{100,000 \thinspace Pa} = 0.01190 \thinspace m^3\] Since all of the Zn reacts with H₂SO₄ and exists only as ZnSO₄ and H₂ gas with the same pressure and temperature, the final volume of the system (V₂) remains the same as V₁. Now, we can calculate the work done by the system: \[\Delta V = V_{2} - V_{1} = 0\] \[w = -P\Delta V = -(100,000 \thinspace Pa)(0 \thinspace m^3) = 0 \thinspace J\] The work (w) for the process is 0 J.

Key concepts

Balanced Chemical Equation

In the context of chemical reactions, a balanced chemical equation ensures that the number of each type of atom on the reactant side is equal to the number on the product side. This stems from the law of conservation of mass, which indicates that matter cannot be created or destroyed.

For the reaction between zinc (Zn) and sulfuric acid (H₂SO₄), the challenge is to write a balanced equation representing each substance involved in the process.
Here, the balanced chemical equation is given by:
\[Zn(s) + H_{2}SO_{4}(aq) \rightarrow ZnSO_{4}(aq) + H_{2}(g)\]

This equation shows that one mole of solid zinc reacts with one mole of aqueous sulfuric acid to produce one mole of zinc sulfate and one mole of hydrogen gas.

• Reactants: Zn and H₂SO₄
• Products: ZnSO₄ and H₂

When balancing chemical equations, it's important to adjust the coefficients — the numbers in front of each molecule — and not change the subscripts of the formulas, as this would change the identity of the substances.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that relates the pressure, volume, temperature, and number of moles of a gas. It's expressed as \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the universal gas constant (8.314 J/mol·K), and \(T\) is the temperature in Kelvin.

In the zinc and sulfuric acid reaction, hydrogen gas is produced. Knowing the number of moles ( \(n = 0.477 \, ext{moles}\)) of hydrogen gas allows us to use the Ideal Gas Law to find the volume it occupies under certain conditions.
Using the equation:
\[ V = \frac{nRT}{P} \]

Given that the pressure \(P = 100,000 \, ext{Pa}\) and the temperature \(T = 300 \, ext{K}\), we can calculate the initial volume provided all these parameters are constant during the reaction.

• This helps us understand gas behavior under the given conditions.
• Remember, it's idealized and not all gases behave perfectly under all conditions.

This simplification allows for easier calculations when predicting the behavior of gas in chemical reactions.

Moles and Energy Calculation

Calculating moles is crucial for understanding chemical reactions and determining the amounts of reactants and products involved. The number of moles \(n\) of a substance is calculated using its mass divided by its molar mass.

For zinc in our reaction:
\[n = \frac{31.2 \, ext{g}}{65.38 \, ext{g/mol}} = 0.477 \, ext{moles} \]

This tells us how much zinc and ultimately how much hydrogen gas will be produced.
Subsequently, the energy change \((\Delta U)\) can be calculated using \(\Delta U = nRT\), where all variables are known.
This energy calculation reflects the internal energy change of the system.

In physical chemistry, such calculations are fundamental for understanding reaction dynamics and energy flow.

• It helps in determining how much energy is needed or released.
• This can influence reaction conditions or the need for catalysts.

Keep in mind, these calculations assume ideal conditions, and real-world deviations might occur.