Question

The atmosphere on Mars is composed mainly of carbon dioxide. The surface temperature is \(220 \mathrm{~K},\) and the atmospheric pressure is about \(6.0 \mathrm{mmHg}\). Taking these values as Martian "STP" calculate the molar volume in liters of an ideal gas on Mars.

Short answer

The molar volume of an ideal gas on Mars is approximately 2288.48 liters.

Step-by-step solution

Convert Pressure to Standard Units

We begin by converting the atmospheric pressure from mmHg to atm. Using the conversion factor, we know that 1 atm = 760 mmHg. Thus, we calculate as follows:\[ \text{Pressure (atm)} = \frac{6.0 \text{ mmHg}}{760 \text{ mmHg/atm}} \approx 0.00789 \text{ atm} \]

Identify Temperature in Kelvin and Constants

Next, we confirm that the temperature is already given in Kelvin: \( T = 220 \text{ K} \). We also use the ideal gas constant \( R = 0.08206 \text{ L atm K}^{-1} \text{ mol}^{-1} \).

Apply the Ideal Gas Law

Using the ideal gas law \( PV = nRT \) and considering \( n = 1 \text{ mol} \) for calculating molar volume, rearrange the formula to find volume \( V \):\[ V = \frac{nRT}{P} \]

Input the Values into the Formula

Substitute the known values into the formula:\[ V = \frac{(1 \text{ mol}) \times (0.08206 \text{ L atm K}^{-1} \text{ mol}^{-1}) \times (220 \text{ K})}{0.00789 \text{ atm}} \]

Calculate the Molar Volume

Now, perform the calculation:\[ V \approx \frac{18.0532}{0.00789} \approx 2288.48 \text{ L} \]

Key concepts

Molar Volume

Molar volume is a key concept when it comes to gases. It refers to the volume that one mole of a substance occupies. For gases, molar volume is particularly useful because gases fill their containers completely. This means the volume of a gas is directly proportional to the amount of it.
In an ideal gas scenario, this volume is influenced by pressure and temperature as defined by the Ideal Gas Law: \[PV = nRT \] Where:

• \( P \) is the pressure of the gas,
• \( V \) is the volume,
• \( n \) is the number of moles,
• \( R \) is the ideal gas constant,
• \( T \) is the temperature.

The molar volume of a gas is usually calculated at standard conditions of temperature and pressure (STP). However, as seen in the Mars exercise, you can calculate it at different conditions.

Atmospheric Pressure

Atmospheric pressure is the force exerted onto a surface by the weight of the air above that surface. It can differ from one place to another and can be affected by factors such as height above sea level and weather conditions. On Earth, standard atmospheric pressure is defined as 1 atm, equivalent to 760 mmHg. However, the Martian atmosphere is quite different, featuring a low atmospheric pressure around 6 mmHg, primarily because of a thinner atmosphere.
In the example from the Mars exercise, we converted 6 mmHg to atm to maintain consistency with other standard gas law measurements. Proper conversion is crucial as the ideal gas law requires pressure in atmospheres when using the gas constant \( R \) with units of L atm K\(^{-1}\) mol\(^{-1}\).

Standard Temperature and Pressure

Standard Temperature and Pressure (STP) is a reference point used in chemistry to provide a basis for measuring gas volumes. On Earth, STP is set at 0°C (273.15 K) and 1 atm. However, different STP conditions can be used as seen in the Martian example. There, the standard temperature is 220 K and the pressure is about 0.00789 atm.
STP is important because gases behave more predictably under these conditions, allowing for easier application of the Ideal Gas Law to calculate molar volume. Adjusting for particular STP conditions like on Mars helps to correctly assess the behavior of gases under those unique conditions.

Gas Constant

The ideal gas constant, denoted as \( R \), is central to calculations involving gases. It bridges the relationships within the ideal gas law equation; capturing details like energy, temperature, pressure, and volume in a single value. For most calculations involving gases at standard atmospheric pressure measured in atmospheres, its value is \( 0.08206 \, \text{L atm K}^{-1} \text{mol}^{-1} \).
Using \( R \) provides a uniform way to relate these parameters, ensuring that the calculations hold consistent and reliable. Making sure that the same units for \( R \) are used as for the pressure, volume, and temperature values is necessary for accurate results. The consistent use of the gas constant allows for seamless integration across various gas-related calculations.