Question

A hot-air balloon is filled with air to a volume of \(4.00 \times\) \(10^{3} \mathrm{m}^{3}\) at \(745\) torr and \(21^{\circ} \mathrm{C}\). The air in the balloon is then heated to \(62^{\circ} \mathrm{C},\) causing the balloon to expand to a volume of \(4.20 \times 10^{3} \mathrm{m}^{3} .\) What is the ratio of the number of moles of air in the heated balloon to the original number of moles of air in the balloon? (Hint: Openings in the balloon allow air to flow in and out. Thus the pressure in the balloon is always the same as that of the atmosphere.)

Short answer

The ratio of the number of moles of air in the heated balloon to the original number of moles is approximately 0.949.

Step-by-step solution

Identify the given information and convert to SI units

We are given the initial volume (V1), initial temperature (T1), final volume (V2), final temperature (T2), and atmospheric pressure (P), which will let us use the ideal gas law formula nRT = PV. First, we need to convert the pressure to SI units and the temperatures to kelvin. Initial volume: V1 = 4.00 x 10³ m³ Final volume: V2 = 4.20 x 10³ m³ Pressure: P = 745 torr Initial temperature: T1 = 21°C Final temperature: T2 = 62°C Remember that 1 atm = 760 torr, and we also need to convert the pressure to pascals (Pa) for which 1 atm = 101325 Pa. Also, remember that to convert from Celsius to Kelvin, we need to add 273.15 to the Celsius value.

Convert the pressure and temperature

Let's convert the pressure and temperature values to their SI units: P (in atm) = 745 torr × (1 atm / 760 torr) ≈ 0.98 atm P (in Pa) = 0.98 atm × 101325 Pa/atm ≈ 99278 Pa T1 (in K) = 21°C + 273.15 = 294.15 K T2 (in K) = 62°C + 273.15 = 335.15 K Now we have P = 99278 Pa, T1 = 294.15 K, and T2 = 335.15 K.

Use the ideal gas law to relate number of moles and volume

Since the pressure is constant inside and outside the balloon, we can relate the initial and final number of moles (n1 and n2) and the volumes (V1 and V2) using the ideal gas law formula. The universal gas constant (R) is given by R = 8.314 J/(mol K). For the initial state, n1 * R * T1 = P * V1, and for the final state, n2 * R * T2 = P * V2. Divide the first equation by the second equation: (n1 * R * T1) / (n2 * R * T2) = (P * V1) / (P * V2) Because the pressure is constant, we can cancel out R, P, and T: n1 / n2 = (V1 / T1) / (V2 / T2) Now we can find the ratio of the number of moles of air in the heated balloon to the original number of moles.

Calculate the ratio of the number of moles

Plug in the given values for volumes and temperatures into the formula we got in step 3 and calculate the ratio. n1 / n2 = (V1 / T1) / (V2 / T2) = (4.00 x 10³ m³ / 294.15 K) / (4.20 x 10³ m³ / 335.15 K) n1 / n2 ≈ 0.949 The ratio of the number of moles of air in the heated balloon to the original number of moles is approximately 0.949.

Key concepts

Gas Laws

Understanding gas laws is essential to solving problems related to the behavior of gases under different physical conditions. One of the foundational principles is the ideal gas law, which describes how the pressure (P), volume (V), temperature (T), and number of moles (n) of a gas are related under ideal conditions. The ideal gas law is expressed mathematically as \( PV = nRT \), where R is the universal gas constant.

This law proves to be invaluable when determining the state of a gas when any of the other state variables are altered, as seen in the exercise with the hot-air balloon. As long as the gas behaves ideally, which means it follows the kinetic molecular theory, the ideal gas law allows us to predict how changes in volume, temperature, and the number of moles (if the balloon is open) will affect pressure, or vice versa.

Molar Volume

Molar volume is another key concept, particularly when dealing with gases. It refers to the volume occupied by one mole of a substance. In the context of gases, the molar volume at standard temperature and pressure (STP, which is 273.15 K and 1 atm) is \(22.4 L/mol\) for an ideal gas.

In our balloon scenario, understanding molar volume helps explain the relationship between the number of moles of air in the balloon and the balloon's volume at different temperatures. Since the ideal gas law assumes the gas molecules do not interact and take up no volume themselves, variations in real gas behavior may occur at high pressures or low temperatures. However, for our hypothetical balloon and the conditions given, we can assume ideal behavior for a straightforward calculation.

Temperature Conversion

When working with the ideal gas law, it is crucial to convert temperature to an absolute scale. Kelvin (K) is the SI unit for temperature used in gas law equations. To convert Celsius to Kelvin, add 273.15 to the Celsius value.

In our exercise, we are given temperatures in Celsius: T1 at \(21^\circ C\) and T2 at \(62^\circ C\). To properly apply the ideal gas law, we convert these to Kelvin by adding 273.15, giving us T1 as \(294.15 K\) and T2 as \(335.15 K\). This step is fundamental, as temperature directly affects the kinetic energy of the gas particles and, by extension, the pressure and volume.

Pressure Units Conversion

In dealing with gases, pressure measurements come in various units, including atmospheres (atm), torr, and pascals (Pa). It is essential to convert pressures to the same units when conducting calculations. Since the SI unit for pressure is pascals, we often need to convert atm or torr to Pa.

For our exercise, pressure was originally given in torr. To convert it to atm, we used the relationship \(1 atm = 760 torr\). Furthermore, to get the pressure in pascals, we used \(1 atm = 101325 Pa\). These conversions allowed us to use \(99278 Pa\) as the standard pressure in our application of the ideal gas law, ensuring consistency and accuracy in our calculations.