Question

A weather balloon is inflated with helium. The balloon has a volume of \(100 \mathrm{~m}^{3}\) and it must be inflated to a pressure of \(0.10 \mathrm{~atm}\). If \(50 \mathrm{~L}\) gas cylinders of helium at a pressure of \(100 \mathrm{~atm}\) are used, how many cylinders are needed? Assume that the temperature is constant. (a) 2 (b) 3 (c) 4 (d) 1

Short answer

2 cylinders are needed to fill the balloon.

Step-by-step solution

Identify the Given Data

The volume of the balloon is 100 m^3, and it needs to be inflated to a pressure of 0.10 atm. The gas cylinders have a volume of 50 L (which is equivalent to 0.050 m^3) at a pressure of 100 atm.

Apply the Ideal Gas Law

Use the ideal gas law in the form PV = nRT, where P is pressure, V is volume, n is the amount of substance in moles, R is the ideal gas constant, and T is temperature. Since temperature and R are constant, we can use the combined gas law which gives us P1V1 = P2V2, where the subscript 1 refers to the initial condition (cylinder) and the subscript 2 refers to the final condition (balloon).

Calculate the Total Volume Required at Standard Pressure

Find the total volume the helium would occupy at the standard pressure (0.10 atm) if taken from the cylinder. This can be done by rearranging the formula to V2 = (P1V1)/P2. Since the pressure P2 of the balloon (0.10 atm) is lower than the cylinder's pressure P1 (100 atm), the volume of gas needed at the standard pressure will be larger than the volume initially in the cylinder.

Determine the Number of Cylinders Needed

Divide the volume of helium needed at standard pressure (found in the previous step) by the volume of one gas cylinder to obtain the total number of cylinders required to fill the balloon.

Key concepts

The Combined Gas Law

The Combined Gas Law is a fusion of three fundamental gas laws: Boyle's Law, Charles's Law, and Gay-Lussac's Law. It's written as \( P_1V_1\/T_1 = P_2V_2\/T_2 \) where \( P \) stands for pressure, \( V \) for volume, and \( T \) for temperature. This law allows us to calculate changes in pressure, temperature, or volume of a gas when another state changes, as long as the mass and the amount of gas remain constant.

To solve problems using the combined gas law, it is essential to keep units consistent. In this exercise, the temperatures cancel out because it's constant, simplifying the law to \( P_1V_1 = P_2V_2 \) which demonstrates how volume and pressure vary inversely if temperature and the amount of gas remain unchanged. For example, when a gas cylinder inflates a weather balloon, we can predict the final volume of gas needed to achieve the desired pressure by applying this relationship.

Standard Pressure Calculations

Standard pressure is one of the components in the standard conditions for temperature and pressure (STP) used in chemistry to provide a baseline for gas calculations. The commonly accepted value for standard atmospheric pressure is \( 1 atm \) (atmosphere). However, in specific contexts, standard pressure may also be considered as \( 101.325 kPa \) (kilopascals) or \( 760 mmHg \) (millimeters of mercury).

When performing standard pressure calculations, it's crucial to convert all measurements into the same units to maintain accuracy. In gas law problems, you'll often need to convert the real-life pressure conditions to match the standard pressure to calculate the volume or amount of gas at STP. For instance, if we're inflating a balloon to a non-standard pressure like \( 0.10 atm \) from cylinders at \( 100 atm \) we momentarily move our calculations to STP to simplify our problem, as demonstrated in the ideal gas problems.

Stoichiometry in Gases

Stoichiometry is the aspect of chemistry that deals with the quantitative relationships that govern the reactants and products in chemical reactions. In the context of gases, stoichiometry entails using the ideal gas law to connect volumes, pressures, and temperatures to the number of moles of a gas, and further to relate these amounts to other reactants or products in a chemical reaction.

In real-world problem-solving, such as filling a weather balloon with helium, we can use stoichiometry to determine the exact amounts of gas required. Using stoichiometric calculations, one can figure out how many gas cylinders are needed to inflate a balloon - this requires converting volumes at specific pressures and temperatures to moles, and these moles can be compared stoichiometrically to the moles of gas provided by the cylinders.

Chemical Principles for JEE

The Joint Entrance Examination (JEE) is a highly competitive test for admission to engineering colleges in India, and it includes substantial questions on chemical principles. To tackle questions on chemical principles in the JEE, it's imperative to understand gas laws, including the ideal and combined gas laws, principles of stoichiometry, and how substances behave under different temperature and pressure conditions.

In preparation for JEE chemistry, understanding how to efficiently solve ideal gas law problems is advantageous. Students must master the concept of molar volume at STP, Dalton's Law of Partial Pressure, and Graham's Law of Effusion. With these principles, aspiring engineers are well-equipped to manipulate gas variables and apply them to larger, more complex chemical systems, as is often required in the JEE.