Problem 114 - A Complete Resource Book in Chemistry for JEE Main

Question

A gas cylinder has \(370 \mathrm{~g}\) of oxygen at \(298 \mathrm{~K}\) and 30 atm pressure. If the cylinder was heated upto \(348 \mathrm{~K}\) then the valve were held open until the gas pressure was \(1 \mathrm{~atm}\) and the temperature remains \(348 \mathrm{~K}\). What mass of oxygen would escape in this condition? (a) \(349 \mathrm{~g}\) (b) \(359 \mathrm{~g}\) (c) \(329 \mathrm{~g}\) (d) \(339 \mathrm{~g}\)

Step-by-Step Solution

Answer

The mass of oxygen that escaped is 349 g, so the correct answer is (a) 349 g.

Step 1: Calculate Initial Moles of Oxygen

We start by calculating the number of moles of oxygen initially in the cylinder using the ideal gas equation. The ideal gas equation is \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant \(0.0821 \, \text{L atm/mol K}\), and \(T\) is the temperature in Kelvin. We know the initial mass is \(370 \, \text{g}\) and the molar mass of oxygen (\(O_2\)) is \(32 \, \text{g/mol}\). Thus, the initial moles \(n_i = \frac{370}{32}\).

Step 2: Calculate Final Moles of Oxygen

When the valve is opened, the pressure changes while temperature remains constant at \(348 \, K\) and the pressure is reduced to \(1 \, atm\). The volume does not change, so using the fact that initial moles \(n_i = \frac{370}{32}\), apply the combined gas law in the form \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\) to solve for the final moles \(n_f\) at these new conditions. The temperature ratio \(\frac{298}{348}\) is used to find the final number of moles.

Step 3: Determine Mass of Remaining Oxygen

Using the final moles \(n_f\) determined in the previous step, convert this to grams of oxygen remaining in the cylinder using the molar mass of \(O_2\) which is \(32 \, \text{g/mol}\). Calculate: \(\text{mass of remaining oxygen} = n_f \times 32\).

Step 4: Calculate Escaped Oxygen Mass

Subtract the mass of the remaining oxygen from the initial mass to find the mass of oxygen that escaped. \(\text{Escaped mass} = 370 - \text{mass of remaining oxygen}\).

Step 5: Select the Correct Answer

Compare your result for the mass of escaped oxygen to the options provided, and select the closest one.

Key Concepts

Gas Cylinder Problems

When dealing with gas cylinders, understanding how gases behave under different conditions of temperature and pressure is crucial. These problems often involve changes such as heating, cooling, or venting gases. In this scenario, we have a cylinder with a set amount of oxygen at a specific temperature and pressure.
Initially, the cylinder is heated, causing the molecules to move faster, which tends to increase the pressure if the volume remains unchanged. Then, when the cylinder's valve is opened, oxygen escapes until the internal pressure equals the atmospheric pressure. Throughout these changes, understanding the behavior of gases according to the Ideal Gas Law is essential.
Solving such problems requires a systematic approach:

• Identifying initial and final states of the gas - in terms of pressure, volume, and temperature.
• Recognizing changes in state and applying appropriate gas laws.
• Calculating quantities like moles and mass based on these parameters.

By knowing how gases react to changes, you can predict and calculate outcomes effectively, aiding in the safe and efficient handling of gas cylinders.

Moles Calculation

Moles calculation serves as a fundamental step in dealing with gases, which connects mass, volume, pressure, and temperature. To find the number of moles of a gas, we use the Ideal Gas Law, expressed as the equation: \[ PV = nRT \]Here, \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is the temperature. In the given problem, the calculation starts by determining the initial moles of oxygen given the mass, using the molar mass of oxygen, \(32 \text{ g/mol}\). This is found by dividing the mass by the molar mass:\[ n_i = \frac{370}{32} \]Following a change in conditions, recalculation of moles under these new conditions allows us to understand how much gas remains. By rearranging the Ideal Gas Law, we can determine how many moles are present at a lower pressure and different temperature.
Understanding this connection is crucial as it forms the basis for determining other essential properties of gases, such as how much of the gas has escaped when conditions change.

Gas Pressure and Temperature

Gas pressure and temperature are closely linked due to molecular movement in a gas. When temperature changes within a gas cylinder, it directly affects the pressure, following the principles defined by the Ideal Gas Law. In this problem, heating the gas increases molecular speeds, which attempt to increase pressure within the fixed volume of the cylinder.
However, when the valve is opened, oxygen is released until the internal pressure drops to 1 atm, allowing equilibrium with the outside environment. The fixed volume and new conditions set a new baseline, enabling calculation based on the known changes in temperature from 298 K to 348 K.The two states can be connected using the relationship:\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]This helps in determining the final moles and subsequently the mass of oxygen remaining. Understanding these concepts is crucial for managing the dynamics of gases under varying conditions, ensuring accurate assessments and safe use of gas cylinders in practical applications.