Problem 61 - Chemistry and Chemical Reactivity

Question

The temperature of the atmosphere on Mars can be as high as \(27^{\circ} \mathrm{C}\) at the equator at noon, and the atmospheric pressure is about \(8 \mathrm{mm}\) Hg. If a spacecraft could collect 10\. \(\mathrm{m}^{3}\) of this atmosphere, compress it to a small volume, and send it back to Earth, how many moles would the sample contain?

Step-by-Step Solution

Answer

The sample contains about 4.27 moles.

Step 1: Convert Temperature to Kelvin

The given temperature is \(27^{\circ}\mathrm{C}\). To convert this to Kelvin, use the formula: \[ T_K = T_C + 273.15 \] So, \[ T_K = 27 + 273.15 = 300.15\, K \].

Step 2: Convert Pressure to Pascals

The atmospheric pressure on Mars is given as \(8\,\mathrm{mm}\) Hg. We convert this to Pascals using the conversion factor: \(1\,\mathrm{mm}\, Hg = 133.322\, \mathrm{Pa}\). Thus, \[ 8\, \mathrm{mm}\, Hg = 8 \times 133.322 = 1066.576\, \mathrm{Pa} \].

Step 3: Use the Ideal Gas Law

The ideal gas law is \( PV = nRT \), where \(P\) is pressure, \(V\) is volume (in \(m^3\)), \(n\) is moles, \(R\) is the ideal gas constant \(8.314\, \mathrm{J/(mol\cdot K)} \), and \(T\) is temperature in Kelvin. Substitute the known values: \[ \text{Pressure, } P = 1066.576\, \mathrm{Pa}, \, \text{Volume, } V = 10\, \mathrm{m^3}, \, \text{Temperature, } T = 300.15\, K \].

Step 4: Solve for Moles

Rearrange the ideal gas law to solve for \(n\): \[ n = \frac{PV}{RT} \] Substituting the values, \[ n = \frac{1066.576 \times 10}{8.314 \times 300.15} \approx \frac{10665.76}{2497.8631} \approx 4.27 \] moles.

Key Concepts

Mars Atmosphere

Mars, our neighboring planet, has an atmosphere that is quite different from Earth's. It's much thinner and composed mostly of carbon dioxide. This thin atmosphere allows temperatures to fluctuate significantly, especially at the equator, where it can reach up to 27°C at noon. The atmospheric pressure on Mars is also significantly lower compared to Earth, measured in millimeters of mercury (mm Hg). Understanding Mars's atmospheric conditions is crucial for space exploration missions, as these factors affect spacecraft design and mission planning.

Temperature Conversion

To work with temperature in scientific calculations, it's essential to convert degrees Celsius to Kelvin. The Kelvin scale is the standard unit of temperature in scientific studies, as it starts at absolute zero, making it more compatible with other physical laws. The formula for conversion is simple:

• Add 273.15 to the Celsius temperature to convert it to Kelvin.

For example, if the temperature on Mars is 27°C, the conversion to Kelvin would be:

• 27 + 273.15 = 300.15 K.

This conversion is a fundamental step before applying the Ideal Gas Law in calculations.

Pressure Conversion

Atmospheric pressure is typically measured in mm Hg, especially when dealing with astronomical bodies like Mars. To incorporate these measurements into scientific equations, it must be converted to Pascals (Pa), which is the SI unit for pressure. The conversion factor is:

• 1 mm Hg = 133.322 Pa.

For the atmosphere on Mars, which is at 8 mm Hg, the conversion to Pascals can be calculated by multiplying by the conversion factor:

• 8 mm Hg x 133.322 Pa/mm Hg = 1066.576 Pa.

This step ensures measurements are compatible with the ideal gas law calculations for identifying properties like the number of moles.

Moles Calculation

The number of moles in a gas sample is a central part of chemical studies and can be calculated using the Ideal Gas Law, which is represented as:

• \(PV = nRT\)

Where:

• \(P\) is the pressure in Pascals,
• \(V\) is the volume in cubic meters,
• \(R\) is the universal gas constant (8.314 J/(mol·K)),
• \(T\) is the temperature in Kelvin.

By rearranging the equation to solve for \(n\):

• \(n = \frac{PV}{RT}\)

Substituting the values for Mars's atmosphere (1066.576 Pa for pressure, 10 m³ for volume, and 300.15 K for temperature) yields:

• \(n \approx \frac{10665.76}{2497.8631} \approx 4.27\) moles.

This calculation tells us how many moles of the Martian atmosphere are in the sample collected by the spacecraft.