Question

The pressure exerted by a gas in a 10 -lit vessel at \(27^{\circ} \mathrm{C}\) is \(5 \mathrm{~atm}\). IIow many balloons of \(0.5\) litre size can be filled with the outcoming gas? (1) 100 (2) 80 (3) 50 (4) 40

Short answer

The gas can fill 100 balloons of 0.5-liter size.

Step-by-step solution

- Understand Given Data

Given: Volume of the vessel (V1) = 10 liters, Temperature (T) = 27°C, Pressure (P1) = 5 atm, Volume of each balloon = 0.5 liters.

- Convert Temperature to Kelvin

Convert the given temperature from Celsius to Kelvin, using the formula: \[ T (\text{in K}) = 27 + 273.15 = 300.15 \text{ K} \] For simplicity, we approximate it to 300 K.

- Apply Ideal Gas Law

Use the Ideal Gas Law to find the amount of gas in moles: \[ PV = nRT \] Here, R is the gas constant ( R = 0.0821 \text{ L atm } \text{K}^{-1} \text{mol}^{-1})Substituting values, we get: \[ 5 \text{ atm} \times 10 \text{ L} = n \times 0.0821 \times 300 \text{ K} \] Solving for n: \[ n = \frac{5 \times 10}{0.0821 \times 300} \ n \ n \ 2.032 \text{ moles} \]

- Calculate Total Volume of Gas

Using Ideal Gas Law again to find total volume the gas would occupy at 1 atm pressure and 300 K: \[ V = \frac{nRT}{P}\]\[V = \frac{2.032 \times 0.0821 \times 300}{1} \approx 50 \text{ liters}\]

- Calculate Number of Balloons

The volume of each balloon is 0.5 liters. Calculate the number of balloons the gas can fill: \[ \text{Number of balloons} = \frac{50 \text{ liters}}{0.5 \text{ liters}} = 100 \]

- Conclusion

The gas can fill 100 balloons of 0.5-liter size.

Key concepts

Pressure

Pressure is the force exerted by the gas per unit area on the walls of its container. In the Ideal Gas Law, pressure is denoted by the symbol \(P\). In this exercise, we are given that the pressure \(P_1\) inside the vessel is \(5 \, \text{atm}\). Understanding pressure and how it relates to the other variables in the Ideal Gas Law helps us determine the number of moles of gas in the container. Remember, atmospheric pressure is often used in these calculations, hence we measure pressure in units such as atmospheres (atm).

Temperature

Temperature is a measure of the average kinetic energy of the gas particles. In the Ideal Gas Law, temperature must always be in Kelvin (K). To convert Celsius (°C) to Kelvin (K), add 273.15 to the Celsius temperature. For this exercise, the temperature \(T\) is given as 27°C. Using the formula: \[ T \, (\text{in} \,K) = 27 + 273.15 = 300.15 \, K \] For simplicity, we approximate it to 300K. This conversion is crucial because the gas constant \(R\) in the Ideal Gas Law uses Kelvin in its units.

Volume

Volume is the amount of space the gas occupies. In our scenario, the initial volume of the gas is 10 liters. When calculating how many balloons the gas can fill, we consider the volume of each balloon, which is 0.5 liters. The Ideal Gas Law equation \(PV = nRT\) allows us to first calculate the number of moles of the gas and then find the total volume the gas would occupy at standard conditions (1 atm pressure and 300 K). The final volume can then be used to determine how many balloons can be filled.

Mole Calculation

The mole is a basic unit in chemistry representing a quantity of substance. In this exercise, we use the Ideal Gas Law equation \(PV = nRT\) to calculate the number of moles \(n\) of the gas in the vessel. Substituting the given values into the equation, we have: \[ n = \frac{P \times V}{R \times T} \] Replacing \(P = 5 \, \text{atm}\), \(V = 10 \, \text{L}\), \(R = 0.0821 \, \text{L atm} \, \text{K}^{-1} \, \text{mol}^{-1}\), and \(T = 300 \, K\) we get: \[ n = \frac{5 \times 10}{0.0821 \times 300} \ n \ 2.032 \, \text{moles}\] This mole calculation is vital as it forms the basis for determining the total volume of gas available for filling balloons.