Problem 59 The air in a hot-air balloon at ... [FREE SOLUTION] (2024)

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Density is a measure of how much mass is contained within a given volume. It’s usually expressed in grams per liter (g/L) or kilograms per cubic meter (kg/m³).
For a gas, density can be affected by pressure, temperature, and the type of gas itself. In general, increased temperature decreases density - this is why hot air balloons rise when the air inside them is heated!

In calculation, we use the formula: \(\rho = \frac{M \times P}{R \times T}\),
where \(M\) is molar mass, \(P\) is pressure, \(R\) is the gas constant, and \(T\) is temperature.

The ideal gas law is a fundamental equation that describes the relationship between pressure (P), volume (V), temperature (T), and the number of moles (n) of a gas. The formula is: \[ PV = nRT \]
Here:
* \(P\) is the pressure
* \(V\) is the volume
* \(n\) is the number of moles
* \(R\) is the universal gas constant (\(0.0821 \textrm{L} \textrm{​atm} /\textrm{mol} \textrm{​ K}\)
* \(T\) is the temperature in Kelvin

We can rearrange this equation to find density by using the fact that \(n\) (moles) is equal to the mass (m) divided by the molar mass (M): \[ \rho = \frac{MP}{RT} \]
This is exactly what we did in the exercise to find the density at two different temperatures.

In many scientific calculations, the temperature needs to be in Kelvin (K), not Celsius (°C). The conversion formula from Celsius to Kelvin is simple:
\(T(\textrm{K}) = T(\textrm{°C}) + 273.15\)

For example:
* 17°C = 17 + 273.15 = 290.15 K
* 60°C = 60 + 273.15 = 333.15 K
This conversion is crucial because the Kelvin scale is based on absolute zero, the theoretical point where particles have minimum thermal motion.

Pressure is the force exerted per unit area. It can be expressed in different units, including atmospheres (atm), torr, and pascals (Pa).

In the problem, the pressure is given in torr, but we need it in atmospheres for the ideal gas law calculation. The conversion factor is:
\textrm{1 atm} = \textrm{760 torr} \.

Therefore:
744 torr can be converted to atmospheres with:
\(\frac{744 \text{ torr}}{760 \text{ torr/atm}} \approx 0.979 \text{ atm} \)
This helps standardize measurements, making calculations easier.

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol).

For air, the average molar mass is given as 28.8 g/mol.
This value represents the weighted average of the molar masses of the different gases that make up air, such as nitrogen, oxygen, and argon.

The molar mass is used in the ideal gas law to calculate other properties like density.
We can apply it as:
\( \rho = \frac{M \times P} {R \times T}\),
where Molar Mass (M) is essential for determining the mass in this context.