Problem 63 - 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 approximately 4.27 moles of gas.

Step 1: Convert Temperature to Kelvin

The temperature on Mars is given as \(27^{\circ}C\). To use the ideal gas law, we need to convert this to Kelvin. The formula for converting Celsius to Kelvin is \(T_{K} = T_{C} + 273.15\). So,\[T_{K} = 27 + 273.15 = 300.15\, K.\]

Step 2: Convert Pressure to Pascals

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

Step 3: Apply the Ideal Gas Law

The ideal gas law is \(PV = nRT\), where \(R\) is the ideal gas constant \(8.314\, \mathrm{J} / \mathrm{mol}\cdot \mathrm{K}\). We need to find the number of moles \(n\). Given \(V = 10\, \mathrm{m}^{3},\) \(P = 1066.576\, \mathrm{Pa}\), and \(T = 300.15\, \mathrm{K}\):\[n = \frac{PV}{RT} = \frac{1066.576 \times 10}{8.314 \times 300.15}.\]

Step 4: Calculate the Number of Moles

Continuing from the previous step, we substitute the values into the equation.\[n = \frac{10665.76}{2495.5621} \approx 4.27\, \mathrm{moles}.\]

Key Concepts

Temperature Conversion

Understanding temperature conversion is crucial when dealing with gas laws. The Ideal Gas Law makes use of the Kelvin scale because it begins at absolute zero—an essential requirement for thermodynamic equations. To convert Celsius, the scale most commonly used, to Kelvin, you simply add 273.15.

If the temperature on Mars is given as 27°C, using the formula:

• Convert to Kelvin: \[ T_{K} = T_{C} + 273.15 = 27 + 273.15 = 300.15 \, K \]

This conversion ensures that all calculations using the gas laws are performed with consistent units. Kelvin provides a uniform baseline that aligns with the physical laws governing heat and energy.

Pressure Conversion

Pressure is also a key variable in the Ideal Gas Law and must be converted into standard units for the calculation. Atmospheric pressure on Mars is given in millimeters of mercury (mmHg), a unit often used because of its relation to barometric pressure. However, in scientific equations, like the Ideal Gas Law, pressure is usually expressed in Pascals (Pa).

The conversion factor for millimeters of mercury to Pascals is:

• \[1 \, \text{mmHg} = 133.322 \, \text{Pa} \]
• Mars' atmospheric pressure: \[8 \, \text{mmHg} = 8 \times 133.322 = 1066.576 \, \text{Pa} \]

This conversion allows consistent usage of units across physical and chemical equations, ensuring accuracy in results.

Molecular Calculations

The Ideal Gas Law \(PV = nRT\) allows us to find the number of moles in a sample. Here,  \(n\) represents the amount of substance (moles),  \(P\) is the pressure,  \(V\) is the volume,  \(R\) is the ideal gas constant, and  \(T\) is the temperature in Kelvin.

To find the number of moles, rearrange the equation:

• \[n = \frac{PV}{RT}\]
• Substitute known values: \[n = \frac{1066.576 \times 10}{8.314 \times 300.15}\]
• Perform the calculation: \[n = \frac{10665.76}{2495.5621} \approx 4.27 \, \text{moles}\]

By using these steps, you can determine the amount of gas moles collected, crucial for understanding the composition of Mars' atmosphere.

Ideal Gas Constant

The Ideal Gas Constant (denoted as \(R\)) is a factor in the Ideal Gas Law that connects pressure, volume, temperature, and amount of gas. It has a fixed value of 8.314 J/mol·K in SI units, enabling the application of the gas law under ideal conditions.

In the context of gas laws:

• The constant \(R\) remains unchanged, allowing for straightforward substitutions once the other variables are known.
• It bridges the relationships between different units in scientific calculations, ensuring consistency and accuracy.

The constancy of \(R\) simplifies computations across varied gas samples and conditions, making it indispensable in these analyses.