Problem 76 - Chemistry: An Atoms-Focused Approach

Question

A balloon vendor at a street fair is using a tank of helium to fill her balloons. The tank has an internal volume of \(45.0 \mathrm{L}\) and a pressure of 195 atm at \(22^{\circ} \mathrm{C}\). After a while she notices that the valve has not been closed properly and the pressure has dropped to 115 atm. How many moles of He have been lost?

Step-by-Step Solution

Answer

The given problem asks to calculate the amount of helium gas, in moles, that have been lost from the helium tank. By applying the Ideal Gas Law, we find the initial amount of helium gas to be approximately 449.924 moles. After the loss of some gas that resulted in a drop in pressure, the final amount of helium gas was calculated to be approximately 282.071 moles. The difference between the initial and final amount, which is roughly 167.853 moles, represents the amount of helium gas that has been lost.

Step 1: Write down the Ideal Gas Law formula

The Ideal Gas Law can be expressed as follows: \(PV=nRT\) where P is the pressure (in atm), V is the volume (in L), n is the number of moles of the gas, R is the Ideal Gas constant (= 0.08206 L atm/mol K), and T is the temperature (in Kelvin).

Step 2: Convert the given temperature to Kelvin

To convert the temperature from Celsius to Kelvin, we can use the formula: \(T(K) = T(^{\circ}C) + 273.15\) So, the temperature in Kelvin is: \(T(K) = 22 + 273.15 = 295.15 K\)

Step 3: Calculate the initial number of moles of helium gas

Using the Ideal Gas Law formula, we can calculate the initial number of moles (n) of helium gas in the tank when the pressure is 195 atm: \(PV = nRT\) Rearrange to solve for n: \(n = \frac{PV}{RT}\) Plug in the given values: \(n = \frac{(195\,\text{atm})(45.0\,\text{L})}{(0.08206\,\text{L atm/mol K})(295.15\,\text{K})}\) \(n \approx 449.924\,\text{mol}\) (initial moles of helium gas)

Step 4: Calculate the final number of moles of helium gas

Using the Ideal Gas Law formula again, we can now calculate the final number of moles of helium gas in the tank when the pressure has dropped to 115 atm: \(PV = nRT\) Rearrange to solve for n: \(n = \frac{PV}{RT}\) Plug in the values: \(n = \frac{(115\,\text{atm})(45.0\,\text{L})}{(0.08206\,\text{L atm/mol K})(295.15\,\text{K})}\) \(n \approx 282.071\,\text{mol}\) (final moles of helium gas)

Step 5: Calculate the number of moles of helium gas lost

To find the number of moles of helium gas lost, subtract the final number of moles from the initial number of moles: \(\text{Moles lost} = n_\text{initial} - n_\text{final}\) \(\text{Moles lost} = 449.924\,\text{mol} - 282.071\,\text{mol} \approx 167.853\,\text{mol}\) So, about 167.853 moles of helium gas have been lost from the tank.

Key Concepts

Helium Gas

Helium gas is a noble gas, meaning it is chemically inert and does not easily form compounds with other elements. This makes it perfect for uses like filling balloons, where the gas needs to remain unchanged. Helium is much lighter than air, which is why balloons filled with helium float. Additionally, helium is non-toxic, non-flammable, and has a low boiling point. This makes it safe and ideal for many applications.

• Lighter than air: This property helps balloons to float, creating the desired lifting effect.
• Inert nature: Helium doesn’t react with substances in its environment, preserving the integrity of its state.
• Safety: Being non-flammable and non-toxic makes it safer for public uses.

Understanding these properties is important, as they contribute directly to the behavior of helium under different physical conditions and pressures.

Pressure and Volume

Pressure and volume are two critical components of the Ideal Gas Law, expressed as \(PV = nRT\). In this equation:

• Pressure (P): This is the force exerted by a gas per unit area, measured in atmospheres (atm). Changes in pressure can affect the number of moles of gas.
• Volume (V): This is the space occupied by the gas, measured in liters (L).

When a gas is contained within a fixed volume, like the helium gas in a balloon tank, changes in pressure can result from changes in temperature or the amount of gas. This relationship is crucial for calculating how much gas remains or has been lost during leaks, as described in gas laws.

Temperature Conversion

Temperature conversion is vital in calculations involving gases, as temperature influences gas behavior and properties. The Ideal Gas Law requires that temperature be measured in Kelvin (K). The conversion from Celsius to Kelvin is done through simple addition:

• Convert by adding: To convert Celsius to Kelvin, just add \(273.15\) to the Celsius temperature.

This consistent use of the Kelvin scale is critical because it eliminates negative temperature values that aren't practical for gas calculations. Understanding this conversion process ensures accurate application of the Ideal Gas Law, influencing calculations of moles and behaviors of gases like helium under varying conditions.