Problem 109 - A Complete Resource Book in Chemistry for JEE Main

Question

An L.P.G. cylinder contains \(15 \mathrm{~kg}\) of butane gas at \(27^{\circ} \mathrm{C}\) and 10 atmospheric pressure. It was leaking and its pressure fell down to 8 atmospheric pressure after one day. The gas leaked in 5 days is (a) \(10 \mathrm{~kg}\) (b) \(3 \mathrm{~kg}\) (c) \(15 \mathrm{~kg}\) (d) \(12 \mathrm{~kg}\)

Step-by-Step Solution

Answer

3 kg of gas leaked in one day.

Step 1: Understanding the Problem

The problem states a 15 kg LPG cylinder with butane gas at a given temperature and pressure is leaking. We need to determine how much gas escapes when the pressure reduces from 10 atm to 8 atm in a day.

Step 2: Apply the Ideal Gas Law Before Leak

To understand how much gas was initially present under 10 atm, use the ideal gas law: \(PV = nRT\). Here, we will calculate the initial number of moles using \(P_1 = 10\) atm, \(V\) is constant, \(R\) is the gas constant, and \(T = 27^{\circ}C = 300 K\).

Step 3: Calculate Initial Moles of Gas

15 kg of butane is approximately 15,000 g. The molar mass of butane (C₄H₁₀) is approximately 58 g/mol. Thus, initially, \(n = \frac{15000}{58}\) moles of gas.

Step 4: Apply the Ideal Gas Law After Leak

After the leak, the pressure dropped to 8 atm. Set up the equation \(P_2 = 8\) atm with the same volume and temperature to find the remaining moles: \(n_2\).

Step 5: Calculate Reduced Moles of Gas

The proportion of pressures gives us the proportion of moles remaining: \(\frac{P_2}{P_1} = \frac{n_2}{n_1}\). Thus, \(n_2 = \frac{8}{10} \times n_1\).

Step 6: Determine Moles of Gas Leaked

Subtract the moles remaining from the initial moles to find moles leaked: \(n_{\text{leak}} = n_1 - n_2\).

Step 7: Convert Moles Leaked to Mass

Convert the leaked moles into mass using the molar mass: \(\text{mass}_{\text{leak}} = n_{\text{leak}} \times 58\) g.

Step 8: Calculate Mass Leaked

Using the values calculated: \(n_1 - n_2 = 0.2n_1 = 0.2\times\frac{15000}{58}\). Converting this number of moles to mass, we find the leaking gas weighs 3 kg using the molar conversion of butane.

Key Concepts

LPG Cylinder

LPG, or Liquefied Petroleum Gas, is a common household fuel often found in pressurized metal cylinders for storage and use. Such cylinders store gases like butane, propane, or a mix under pressure in a liquid state. This enables the energy-dense fuel to be conveniently stored and transported. A standard LPG cylinder's main advantage lies in its portability and safety features, designed to handle high-pressure conditions.
The functioning of an LPG cylinder depends on the high pressure inside it. When you open the valve, the gas shifts from high to low pressure, converting from liquid to gas form, available for use in stoves or heaters. Understanding the gas inside, like butane in this case, helps us calculate and predict behaviors like leakage or consumption during use.

Butane Gas

Butane is a hydrocarbon gas and a common constituent of LPG. It consists of carbon and hydrogen atoms, specifically with the formula C₄H₁₀. At room temperature and moderate pressures, butane is typically a gas, but it can be easily liquefied under pressure, making it suitable for storage in LPG cylinders.
As a relatively simple molecule, butane is versatile, often used for cooking, heating, and even as a refrigerant. Its properties make it highly efficient as a fuel, releasing substantial energy when burned. When considering leakage or energy calculations, its molar mass becomes relevant. For butane, this is around 58 g/mol, a critical number for understanding its molecular breakdown and application in calculations involving gas laws.

Gas Leakage Calculation

A critical safety and economic consideration when dealing with LPG cylinders is calculating gas leakage. To quantify how much gas has escaped from a cylinder over time, the Ideal Gas Law becomes instrumental. Consider the change from 10 atm to 8 atm pressure in the LPG cylinder. This implies gas has leaked, reducing the internal pressure. Applying the formula derived from the Ideal Gas Law, you can calculate initial and final moles. Essentially, we express the change as a proportion: the drop from 10 to 8 atm implies a 20% reduction proportionally in gas moles.
Then, convert this percentage into the actual mass of the gas. For instance, if initially there were 15 kg of butane, understanding the moles allows you to convert the 20% loss back into a tangible quantity, like 3 kg of lost gas. Such calculations are vital for efficiently managing resources and ensuring safety.

Moles and Molar Mass

Understanding the concept of moles and molar mass is crucial when dealing with quantities of gases such as butane. A mole is a standard quantity unit in chemistry that represents a large number of molecules or atoms. The molar mass of a substance, like butane’s 58 g/mol, tells us the mass of one mole of its molecules.
For real-world applications, converting gas volumes or masses to moles allows precise calculations with the Ideal Gas Law. In the L.P.G. cylinder example, knowing that 15 kg of butane converts to approximately 258.6 moles (using the conversion 15000g/58 g/mol) provides a foundation to further calculate changes like leakage. This interplay between mass and moles allows for accurate gas management, predicting behavior under changes in volume, pressure, or temperature.

Pressure and Temperature Relationship

The relationship between pressure, volume, and temperature is elegantly encapsulated in the Ideal Gas Law, formulated as \[PV = nRT\]where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles of gas, \(R\) is the gas constant, and \(T\) is temperature in Kelvin.In the context of a cylinder of butane, examine how temperature remains constant if the environment doesn't change significantly. As the pressure drops from 10 atm to 8 atm, assuming volume and temperature stay constant, it indicates a reduction in moles, and therefore mass, within the cylinder.

• The proportionality \(\frac{P_2}{P_1} = \frac{n_2}{n_1}\) simplifies tracking this change.
• A constant temperature ensures that only pressure variations account for changes in moles.

This principle helps predict and manage the behavior of gases under everyday conditions, vital for calculations of gas leakage and safety considerations.