Question

How many moles of methane gas, \(\mathrm{CH}_{4}\), are in a \(100 \mathrm{~m}^{3}\) storage tank at STP? How many grams of methane is this? How many grams of carbon dioxide gas could the same tank hold?

Short answer

4464.29 moles of CH4, 71643 grams of CH4, 196492 grams of CO2.

Step-by-step solution

Understanding STP Conditions

STP (Standard Temperature and Pressure) conditions are defined as 0°C (273.15 K) and 1 atm pressure. At STP, 1 mole of any ideal gas occupies 22.4 liters.

Convert Volume from Cubic Meters to Liters

Since STP conditions give us gas volume in liters, we convert the methane's volume to liters: \(100 \, \text{m}^3 = 100,000 \, \text{liters}\).

Calculate Moles of Methane

Using the volume of one mole of gas at STP (22.4 L), calculate the number of moles of methane: \[ n = \frac{100,000 \, \text{L}}{22.4 \, \text{L/mol}} \approx 4464.29 \, \text{moles}. \]

Convert Moles of Methane to Grams

Find the molar mass of methane (\(\mathrm{CH}_4\)): Carbon (12.01 g/mol) + 4 Hydrogen (4 \times 1.01 g/mol) = 16.05 g/mol. Calculate the mass: \[ \text{Mass} = 4464.29 \, \text{moles} \times 16.05 \, \text{g/mol} \approx 71643 \, \text{grams}. \]

Calculate Mass of Carbon Dioxide in the Same Volume

The molar mass of carbon dioxide (\(\mathrm{CO}_2\)) is calculated as Carbon (12.01 g/mol) + 2 Oxygen (2 \times 16.00 g/mol) = 44.01 g/mol. The number of moles of \(\mathrm{CO}_2\) is still 4464.29 as volume is the same at STP. Therefore, \[ \text{Mass} = 4464.29 \, \text{moles} \times 44.01 \, \text{g/mol} \approx 196492 \, \text{grams}. \]

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that helps to relate the properties of gases under certain conditions. It is given by the formula \( PV = nRT \), where \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin. This formula is essential when working with gases as it allows us to calculate any one of the variables, provided the others are known.

• Pressure (\( P \)) is typically measured in atmospheres (atm) or pascals (Pa).
• Volume (\( V \)) is generally in liters (L) or cubic meters (\( m^3 \)).
• Moles (\( n \)) refer to the amount of substance.
• The gas constant (\( R \)) is typically \( 0.0821 \, \text{L atm/mol K} \).
• Temperature (\( T \)) must always be in Kelvin (K), where \( K = °C + 273.15 \).

In the context of STP, we often use a simplified relationship where a mole of an ideal gas occupies approximately 22.4 L. Knowing the Ideal Gas Law helps us to derive significant information about how gases behave under different conditions.

Molar Mass Calculation

Molar mass is the mass of a given substance (chemical element or chemical compound) divided by its amount of substance, measured in moles. It is crucial for converting between grams and moles, which is a common requirement in stoichiometry. To calculate the molar mass:

• Identify each element in the molecular formula.
• Use the periodic table to find the atomic mass of each element.
• Multiply the atomic mass by the number of times the element appears in the formula.
• Add up all these values to get the molar mass of the compound.

For instance, the molar mass of methane (\( \mathrm{CH}_4 \)):

• Carbon is \( 12.01 \) g/mol.
• Each hydrogen atom is \( 1.01 \) g/mol.
• Total is \( 12.01 + (4 \times 1.01) = 16.05 \) g/mol.

Molar mass is a useful concept for quickly converting between the mass of an element or compound that you have and the moles or particles that it contains, facilitating various calculations in chemical reactions.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure, or STP, is a set of conditions for experimental measurements to enable comparisons between different sets of data. It specifically defines the temperature as 0°C (273.15 K) and the pressure as 100 kPa or 1 atm. STP is often used when dealing with gases since many properties behave predictably under these conditions. At these conditions, any ideal gas occupies a volume of 22.4 liters per mole, known as the molar volume.

• This provides a simple reference for calculating gas volumes or determining quantities when conditions meet STP.
• It's a standard reference point, ensuring consistency across experiments and educational scenarios.

Understanding STP is essential for problems related to gas volumes, especially when using the Ideal Gas Law, as it simplifies calculations by providing a known baseline for temperature and pressure.