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 of air in the balloon is approximately 0.921.

Step-by-step solution

Convert temperatures to Kelvin

To use the ideal gas law, the temperatures should be in Kelvin. Convert the Celsius temperatures to Kelvin by adding 273.15. \(T1_{Kelvin} = T1_{Celsius} + 273.15 = 21 + 273.15 = 294.15~K\) \(T2_{Kelvin} = T2_{Celsius} + 273.15 = 62 + 273.15 = 335.15~K\)

Relate volume and temperature using ideal gas law

Since we have constant pressure, let's express the ideal gas law relating the initial and final conditions, taking the ratio n2/n1: \(\frac{n2}{n1} = \frac{P2V2T1}{P1V1T2}\) As P1 = P2, they can be canceled out: \(\frac{n2}{n1} = \frac{V2T1}{V1T2}\)

Substitute the given values

Now, substitute the given values for V1, V2, T1, and T2: \(\frac{n2}{n1} = \frac{(4.20\times10^{3}~m^3)(294.15~K)}{(4.00\times10^{3}~m^3)(335.15~K)}\)

Calculate the ratio

Perform the calculations: \(\frac{n2}{n1} = \frac{1.236\times10^{6} ~m^3K}{1.342\times10^{6} ~m^3K}\) \(\frac{n2}{n1} ≈ 0.921\) The ratio of the number of moles of air in the heated balloon to the original number of moles of air in the balloon is approximately 0.921.

Key concepts

Gas Laws

The Ideal Gas Law is a fundamental equation in chemistry and physics that relates pressure, volume, and temperature of a gas to the number of molecules present. - The equation is expressed as \(PV = nRT\), where: - \(P\) is the pressure, - \(V\) is the volume, - \(n\) is the number of moles, - \(R\) is the universal gas constant, - \(T\) is the temperature in Kelvin.In scenarios where the pressure is constant, as in the case of a hot-air balloon with openings that allow air to enter or escape to maintain atmospheric pressure, the relationship simplifies. The gas laws, when applied correctly, can help us understand how a change in volume is related to temperature and amount of gas. This is specifically useful when heating air in a balloon, prompting volume expansion.

Temperature Conversion

Temperature conversion is essential when working with the Ideal Gas Law because the law requires temperatures to be in Kelvin. Here’s why:- **Kelvin Scale**: Begins at absolute zero, making it ideal for scientific calculations.- **Conversion Formula**: \[ T_{Kelvin} = T_{Celsius} + 273.15 \]For example:- Converting from Celsius to Kelvin for a temperature of \(21^{ ext{C}}\): \[ 21 + 273.15 = 294.15~K \]- Converting from Celsius to Kelvin for a temperature of \(62^{ ext{C}}\): \[ 62 + 273.15 = 335.15~K \]Always remember to convert to Kelvin to prevent errors in calculations involving gas laws. This ensures that the physical conditions scale correctly with the properties being measured.

Mole Ratio

The mole ratio in gas laws tells us how the number of molecules (or moles) changes with varying conditions. In the Ideal Gas Law, the mole ratio of a gas can be determined by changes in pressure, volume, and temperature as expressed in the simplified form:- With constant pressure, the ratio of moles is: \[ \frac{n2}{n1} = \frac{V2T1}{V1T2} \] - \(n1\) and \(n2\) are the initial and final moles, - \(V1\) and \(V2\) are the initial and final volumes, - \(T1\) and \(T2\) are the initial and final temperatures.In our balloon example, after heating, the calculations yielded a mole ratio of \(0.921\), indicating the moles of heated air are about 92.1% of the original moles, due to some air escaping through the openings as the balloon expands.

Volume Expansion

Volume expansion refers to how the volume of a gas increases with a rise in temperature, provided the pressure remains constant. This concept is particularly evident in gases, which are highly expansible compared to solids and liquids. - **Thermal Expansion**: As gases are heated, their molecules move more quickly, causing the gas to expand. This is a direct application of Charles' Law, which states: - The volume of a gas is directly proportional to its temperature (in Kelvin) when pressure is constant.- **Experiment Consideration**: - Given a balloon with a volume change from \(4.00 \times 10^{3}~m^{3}\) to \(4.20 \times 10^{3}~m^{3}\), this increase (5%) highlights thermal expansion effects.Understanding volume expansion is pivotal in applications like hot-air balloons, where managing these physical changes is crucial for ensuring buoyancy and stability.