Question

A 15.0 -L tank is filled with \(\mathrm{H}_{2}\) to a pressure of \(2.00 \times 10^{2} \mathrm{atm}\) . How many balloons (each 2.00 \(\mathrm{L}\) ) can be inflated to a pressure of 1.00 \(\mathrm{atm}\) from the tank? Assume that there is no temperature change and that the tank cannot be emptied below 1.00 atm pressure.

Short answer

1492 balloons can be inflated to a pressure of 1.00 atm using the gas from the tank.

Step-by-step solution

Write down the initial conditions of the gas in the tank

The initial conditions of the gas in the tank are as follows: - Volume (V1): 15.0 L - Pressure (P1): \(2.00 \times 10^2\) atm Additionally, we should note that the tank cannot be emptied below 1.00 atm pressure.

Write down the ideal gas law equation for the given conditions

In this step, we will use the ideal gas law equation which states that for any gas, \(P_1V_1 = P_2V_2\), where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume, respectively.

Calculate the final conditions of the gas in the tank

As mentioned, we cannot empty the tank below 1.00 atm pressure. Therefore, we have: - Final Pressure (P2): 1.00 atm Now, we can re-arrange the ideal gas law equation to find the final volume (V2): \(V_2 = \frac{P_1V_1}{P_2}\) Substitute the given values, and find V2: \(V_2 = \frac{(2.00 \times 10^2 \mathrm{atm})(15.0 \mathrm{L})}{1.00 \mathrm{atm}} = 3000 \mathrm{L}\)

Calculate the volume of gas available for inflating the balloons

Now, find the volume available for inflating the balloons by taking the difference between V2 and the initial volume of the tank (V1): \(V_\mathrm{available} = V_2 - V_1 = 3000 \mathrm{L} - 15.0 \mathrm{L} = 2985 \mathrm{L}\)

Divide the available volume of the gas by the volume of one balloon

Since each balloon needs to be inflated to a volume of 2.00 L at a pressure of 1.00 atm, we can find the total number of balloons: \(N_\mathrm{balloons} = \frac{V_\mathrm{available}}{V_\mathrm{balloon}}\) Substitute the values and find the number of balloons: \(N_\mathrm{balloons} = \frac{2985 \mathrm{L}}{2.00 \mathrm{L}} = 1492.5\) Since we cannot have a fraction of a balloon, we have to round down the number of balloons to the nearest whole number. Therefore, the final answer is:

Final answer

1492 balloons can be inflated to a pressure of 1.00 atm using the gas from the tank.

Key concepts

Gas Pressure

Gas pressure is the force applied by gas molecules when they collide with the walls of their container. This occurs because the molecules of a gas are always in motion. When they hit the walls of a container, they exert a force that we perceive as pressure.
In the problem, the tank holds hydrogen gas at an initial pressure of \(2.00 \times 10^2\) atm. This pressure indicates the force exerted by the gas molecules per unit area on the walls of the 15.0 L tank. As we use the gas to inflate balloons, the pressure reduces since we are transferring gas molecules into larger volumes, increasing the space for the gas to spread out.

Gas Volume

Gas volume refers to the amount of space that the gas occupies. For an ideal gas, the volume can be altered by changing pressure or temperature, due to the relationship outlined by the Ideal Gas Law.
In our exercise, the volume of the tank is set at 15.0 L initially, but after utilizing the gas according to the Ideal Gas Law, it effectively expands to fill a volume of 3000 L at a reduced pressure of 1.00 atm. This highlights how gas volume can dramatically increase when pressure decreases, assuming a constant temperature.

Balloon Inflation

Balloon inflation involves filling a balloon with gas to a volume where it reaches a certain pressure. In this case, each balloon is inflated to 2.00 L, with the pressure set at 1.00 atm. The relation of volume and pressure during inflation is constant as per the conditions specified; no change in temperature (Ideal Gas Law).
When you start inflating, gas exits the tank at a higher pressure and reduces in pressure as it occupies the larger balloon volume. This process continues until you achieve equilibrium with the external atmospheric pressure, thus maintaining the inflated balloon at 1.00 atm.

Pressure-Volume Relationship

The pressure-volume relationship is vital in understanding how gas behaves under different conditions. According to Boyle’s Law, an aspect of the Ideal Gas Law, the pressure of a gas is inversely proportional to its volume, as long as the temperature remains constant.
This concept can be seen when transferring gas from the tank into balloons. Initially, the gas is under high pressure in a small volume (15.0 L in the tank). As the gas is released, it fills a larger total volume (3000 L), which causes the pressure to drop to 1.00 atm. This change enables us to inflate as many as 1492 balloons, each with a volume of 2.00 L at that same pressure, illustrating this inverse relationship between pressure and volume.