Problem 157 - A Complete Resource Book in Chemistry for JEE Main

Question

Hydrogenation of vegetable ghee at \(27^{\circ} \mathrm{C}\) reduces the pressure of \(\mathrm{H}_{2}\) from \(3 \mathrm{~atm}\) to \(2.18 \mathrm{~atm}\) in 40 minutes. The rate of reaction in terms of molarity per second is \(\left(\mathrm{R}=0.082 \mathrm{~L} \mathrm{~atm} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right)\) (a) \(1.357 \times 10^{-6}\) (b) \(1.537 \times 10^{-5}\) (c) \(1.375 \times 10^{-5}\) (d) \(6.250 \times 10^{-4}\)

Step-by-Step Solution

Answer

The correct rate of reaction is \(1.375 \times 10^{-5}\) M/s, option (c).

Step 1: Convert Temperature to Kelvin

First, convert the temperature from Celsius to Kelvin by adding 273 to the Celsius temperature.\[ T = 27 + 273 = 300 \, \text{K} \]

Step 2: Calculate Change in Pressure

Determine the change in pressure of \( \text{H}_2 \) by subtracting the final pressure from the initial pressure.\[ \Delta P = P_{\text{initial}} - P_{\text{final}} = 3.0 \, \text{atm} - 2.18 \, \text{atm} = 0.82 \, \text{atm} \]

Step 3: Determine Volume Using Ideal Gas Law

Use the Ideal Gas Law to estimate the volume using an arbitrary amount, since the volume will cancel out in rate calculations. Assume \( n = 1 \, \text{mol} \) to simplify:\[ P V = n R T \]\[ 3 \, \text{atm} \cdot V = 1 \, \text{mol} \cdot 0.082 \, \text{L atm mol}^{-1} \text{K}^{-1} \cdot 300 \, \text{K} \]Rearranging gives:\[ V = \frac{24.6 \, \text{L atm}}{3 \, \text{atm}} = 8.2 \, \text{L} \]

Step 4: Calculate Change in Moles Using PV=nRT

Determine the change in moles using the Ideal Gas Law \( PV = nRT\):\[ \Delta n = \frac{\Delta P \times V}{R \times T} \]\[ \Delta n = \frac{0.82 \, \text{atm} \times 8.2 \, \text{L}}{0.082 \, \text{L atm mol}^{-1} \text{K}^{-1} \times 300 \, \text{K}} \approx 0.273 \text{ mol} \]

Step 5: Convert Time for Rate Calculation

Convert the time from minutes to seconds for rate calculation.\[ 40 \, \text{minutes} = 40 \times 60 = 2400 \, \text{seconds} \]

Step 6: Calculate Rate of Reaction

Use the change in moles and time to find the rate of reaction in moles per liter per second.Given that volume \( V = 8.2\, \text{L} \),\[ \text{Rate} = \frac{0.273 \, \text{mol}}{8.2 \, \text{L} \times 2400 \, \text{s}} = 1.375 \times 10^{-5} \, \text{M/s} \]

Step 7: Select Correct Option

Compare the calculated rate to the provided options and select the correct one.- (a) \(1.357 \times 10^{-6}\)- (b) \(1.537 \times 10^{-5}\)- (c) \(1.375 \times 10^{-5}\)- (d) \(6.250 \times 10^{-4}\)The correct answer is: (c) \(1.375 \times 10^{-5}\).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that relates pressure, volume, temperature, and the number of moles of a gas. It is expressed as \( PV = nRT \), where:

• \( P \) is the pressure of the gas in atmospheres (atm).
• \( V \) is the volume in liters (L).
• \( n \) is the number of moles of the gas.
• \( R \) is the Universal Gas Constant, with a value of \( 0.082 \, \text{L atm mol}^{-1} \text{K}^{-1} \).
• \( T \) is the temperature in Kelvin (K).

You can use this law to find any unknown property if the others are known. In this exercise, we assumed \( n = 1 \) mol to simplify calculations, allowing us to determine the volume based on initial pressure and temperature. When working with reactions involving gases, this law is crucial for predicting how changes in conditions affect the reaction.

Pressure Change

Pressure change is critical in understanding reactions involving gases. It's calculated by the difference between initial and final pressures: \[ \Delta P = P_{\text{initial}} - P_{\text{final}} \]In the hydrogenation process mentioned here, pressure decreased from \(3 \, \text{atm} \) to \(2.18 \, \text{atm}\), resulting in a change of \(0.82 \, \text{atm}\). This decrease indicates the consumption of hydrogen gas during the reaction.
The change in pressure directly impacts the gas volume and moles, as dictated by the Ideal Gas Law. Monitoring pressure variations is essential for calculating the reaction rate, offering insight into the reaction's pace and completeness.

Temperature Conversion

When using the Ideal Gas Law, temperatures must be in Kelvin. Converting Celsius to Kelvin requires adding 273 to the Celsius temperature:\[ T = \text{Temperature in °C} + 273 \]In the exercise, the temperature conversion from \(27^{\circ} \text{C} \) results in \(300 \, \text{K}\).Using Kelvin avoids zero or negative values, which are incompatible with the laws of thermodynamics and calculations involving absolute temperature scales. Accurate temperature conversion is essential to predict gas behavior correctly under different conditions.

Reaction Rate

The reaction rate indicates how fast a chemical reaction occurs, usually expressed in terms of concentration change over time. For this exercise, it's given as molarity per second (M/s). Steps to calculate reaction rate include:

• Determine the change in moles of reactant: here it was \(0.273\) moles for hydrogen.
• Convert the reaction time to seconds, crucial for accurately capturing how fast the rate is—2400 seconds in the example.
• Use the formula: \[ \text{Rate} = \frac{\Delta n}{V \times \Delta t} \]

Applied in this situation, knowing the change in moles and total volume gives us the reaction rate of \(1.375 \times 10^{-5} \, \text{M/s}\), revealing the speed at which hydrogen is consumed during hydrogenation.

Molarity

Molarity is the measure of the concentration of a solution, expressed as moles of solute per liter of solution (M). It's a key factor when calculating reaction rates in a solution-based reaction.

• In reactions involving gases, this relates to the volume in which gas reactions take place, translating pressure changes into concentration.
• Molarity enables us to express the reaction rate, with units typically presented as M/s.

In this exercise, molarity was calculated by dividing the change in moles by the total reaction volume and time. Understanding molarity allows us to assess the extent of reaction and adjust conditions to optimize the process.