Problem 70 - Chemistry The Science in Context

Question

A balloon vendor at a street fair is using a tank of helium to fill her balloons. The tank has a volume of \(145 \mathrm{L}\) and a pressure of 136 atm at \(25^{\circ} \mathrm{C}\). After a while she notices that the valve has not been closed properly, and the pressure has dropped to 94 atm. How many moles of gas have been lost?

Step-by-Step Solution

Answer

Answer: Approximately 209.4 moles of gas have been lost.

Step 1: Convert temperature to Kelvin and use appropriate R value

First, we need to convert the given temperature from Celsius to Kelvin and use the appropriate R value (0.0821 L atm/mol K) for our calculations. Given T in Celsius: \(T = 25^{\circ} C \) To convert it to Kelvin, add 273.15: \(T = 25 + 273.15 = 298.15 K\)

Step 2: Calculate the initial number of moles(n1) using the Ideal Gas Law

We have the following initial values: \(V_1 = 145 L\) \(P_1 = 136 atm\) \(T_1 = 298.15 K\) \(R = 0.0821 L atm/mol K\) Now, we plug these values into the Ideal Gas Law and solve for n1: \(P_1V_1 = n_1RT_1\) \( \Rightarrow n_1 = \frac{P_1V_1}{RT_1}\) \( \Rightarrow n_1 = \frac{136 * 145}{0.0821 * 298.15} \approx 676.9 \) So, initially, there are approximately 676.9 moles of gas in the tank.

Step 3: Calculate the final number of moles(n2) using the Ideal Gas Law

Now, we have the following final values: \(V_2 = 145 L\) (volume remains the same) \(P_2 = 94 atm\) \(T_2 = 298.15 K\) (temperature remains the same) \(R = 0.0821 L atm/mol K\) Applying the Ideal Gas Law again to find n2: \(P_2V_2 = n_2RT_2\) \(\Rightarrow n_2 = \frac{P_2V_2}{RT_2}\) \(\Rightarrow n_2 = \frac{94 * 145}{0.0821 * 298.15} \approx 467.5\) So, there are approximately 467.5 moles of gas remaining in the tank.

Step 4: Calculate the number of moles of gas lost

Finally, we find the difference between the initial and final moles to determine the number of moles of gas lost: \(\Delta n = n_1 - n_2 = 676.9 - 467.5 \approx 209.4\) Hence, approximately 209.4 moles of gas have been lost.

Key Concepts

Moles of Gas

The concept of moles is fundamental in chemistry and is particularly important when dealing with gases. A mole is a unit used to measure the amount of a substance. It is based on the number of atoms in 12 grams of carbon-12, which is approximately 6.022 x 10^23 particles (known as Avogadro's number).

When we use the Ideal Gas Law, moles help us understand the quantity of gas present in a container under certain conditions. The Ideal Gas Law is given by the formula:\[PV = nRT\]where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume of the gas.
• \(n\) is the number of moles.
• \(R\) is the ideal gas constant.
• \(T\) is the temperature in Kelvin.

In our problem, moles of gas lost from the tank are calculated by finding the difference in moles before and after the pressure drop. This difference tells us how many moles leaked out when the valve was not properly closed.

Temperature Conversion

When dealing with gas laws, it is important to express the temperature in Kelvin. Kelvin is the SI unit for temperature and is used because it is an absolute scale that starts at absolute zero, consistent across all scientific calculations.

To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature. For instance, to convert \(25^{\circ} C\) to Kelvin:\[T_{Kelvin} = 25 + 273.15 = 298.15 \text{ K}\].This ensures that the temperature units match the ideal gas constant \(R\) employed in the equation \(PV = nRT\). Failing to convert temperature can lead to errors in calculating the amount of gas, as temperature greatly influences how gases behave under different conditions.

Gas Pressure

Gas pressure is the force exerted by gas particles colliding with the walls of their container. It is typically measured in atmospheres (atm), a convenient unit when using the Ideal Gas Law since the constant \(R = 0.0821 \ \text{L atm/mol K}\) is based on these units.

In our scenario, the initial pressure is 136 atm, which then falls to 94 atm after the valve is improperly closed. This change in pressure affects the number of moles of gas present in the tank, as seen when applying the Ideal Gas Law.

Accurately knowing the gas pressure is vital for calculating the amount of gas in a container, predicting how the gas will behave when conditions change, and ensuring safe handling in practical applications.

Volume of Gas

The volume of a gas is the amount of space that the gas occupies, usually measured in liters (L). It's a crucial factor in calculations involving gas laws. In the Ideal Gas Law, volume is directly proportional to the amount of gas (moles) and temperature, while inversely proportional to pressure when using the formula \(PV = nRT\).

In the problem scenario, the volume of the helium tank is constant at 145 L. This consistency allows us to focus on the changes in pressure and moles because they directly influence the calculations without changing the overall context of the tank's capacity. The volume being constant simplifies calculations and helps isolate the effects of temperature and pressure changes on the quantity of gas.