Problem 94 - A Complete Resource Book in Chemistry for JEE Main

Question

One litre of gas \(\mathrm{A}\) at \(2 \mathrm{~atm}\) pressure at \(27^{\circ} \mathrm{C}\) and two litres of gas \(\mathrm{B}\) at \(3 \mathrm{~atm}\) pressure at \(127^{\circ} \mathrm{C}\) are mixed in a 4 litre vessel. The temperature of the mixture is maintained at \(327^{\circ} \mathrm{C}\). What is the total pressure of the gaseous mixture? (a) \(3.93 \mathrm{~atm}\) (b) \(3.25 \mathrm{~atm}\) (c) \(4.25 \mathrm{~atm}\) (d) \(6.25 \mathrm{~atm}\)

Step-by-Step Solution

Answer

The total pressure of the gaseous mixture is 3.25 atm.

Step 1: Identify Initial Conditions for Gas A and Gas B

Gas A has an initial pressure of 2 atm, volume of 1 L, and temperature of 27°C. Gas B has an initial pressure of 3 atm, volume of 2 L, and temperature of 127°C.

Step 2: Convert Temperatures to Kelvin

Convert the temperatures from Celsius to Kelvin using the formula: \[ T(K) = T(°C) + 273.15 \] For Gas A, \(T_A = 27 + 273.15 = 300.15\) K. For Gas B, \(T_B = 127 + 273.15 = 400.15\) K.

Step 3: Apply Ideal Gas Law to Find Moles of Each Gas

We use the ideal gas law in the form \( PV = nRT \) to solve for \( n \), the number of moles:For gas A:\[ n_A = \frac{P_AV_A}{RT_A} = \frac{2 \times 1}{0.0821 \times 300.15} \approx 0.0811 \text{ moles} \]For gas B:\[ n_B = \frac{P_BV_B}{RT_B} = \frac{3 \times 2}{0.0821 \times 400.15} \approx 0.1836 \text{ moles} \]

Step 4: Calculate Total Moles of the Mixture

The total number of moles in the mixture is:\[ n_{\text{total}} = n_A + n_B = 0.0811 + 0.1836 = 0.2647 \text{ moles} \]

Step 5: Use Ideal Gas Law to Find Total Pressure in New Conditions

The temperature of the mixture is 327°C, which is:\[ T_{\text{mixture}} = 327 + 273.15 = 600.15 \text{ K} \]The volume of the vessel is 4 L, so using the ideal gas law:\[ P_{\text{total}} \times V = n_{\text{total}} \times R \times T_{\text{mixture}} \]Solving for the total pressure:\[ P_{\text{total}} = \frac{n_{\text{total}} \times R \times T_{\text{mixture}}}{V} = \frac{0.2647 \times 0.0821 \times 600.15}{4} \approx 3.25 \text{ atm} \]

Key Concepts

Gas Mixtures

When dealing with gas mixtures, it's essential to consider each gas's initial conditions before they mix. Each gas behaves independently according to the ideal gas law. When mixed, they occupy a common volume, but their interactions with the walls of the container dictate the total pressure. A gas mixture's behavior can still be described using the ideal gas law as long as you account for the total moles of all gases involved. In our problem, gases A and B mix in a 4-liter vessel after starting in separate volumes. Each has its unique temperature and volume, initially, and upon mixing, the gases equilibrate to the same temperature and volume.

Pressure Calculation

Pressure calculation in gas mixtures involves figuring out how the individual gases' pressures contribute to the total pressure after mixing. In mixed gases, the total pressure is the sum of the partial pressures of each gas, assuming no interaction between them, following Dalton's Law. Here, we computed pressure using the ideal gas law: the total number of moles and the new temperature and volume conditions were essential. We calculated the final pressure to ensure it reflects the conditions after mixing all gases together at a new uniform temperature.

Temperature Conversion

Temperature conversion is pivotal in gas calculations since the ideal gas law requires Kelvin as the temperature unit. This is because Kelvin uses absolute zero as the baseline, providing consistency in calculations. To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature. For example, converting 27°C to Kelvin gives 300.15 K. Always ensure temperature is in Kelvin before inserting it into formulas dealing with gas calculations, ensuring accurate results.

Mole Calculation

Mole calculation is an integral part of applying the ideal gas law. The number of moles, denoted as \( n \), is found by rearranging the ideal gas law formula: \( n = \frac{PV}{RT} \). To solve this equation, you need the gas's pressure \( P \), volume \( V \), temperature \( T \), and the ideal gas constant \( R \), which is 0.0821 L·atm/(K·mol). In our example, calculating moles involved determining \( n_A \) and \( n_B \) independently using their respective pressures, volumes, and temperatures before the gases were mixed.