Question

Weight of \(\mathrm{CO}_{2}\) in a \(10 \mathrm{~L}\) cylinder at 5 atm and \(27^{\circ} \mathrm{C}\) is (a) \(200 \mathrm{~g}\) (b) \(224 \mathrm{~g}\) (c) \(44 \mathrm{~g}\) (d) \(89.3 \mathrm{~g}\)

Short answer

The weight of CO2 in the 10 L cylinder is 200 g.

Step-by-step solution

Convert Temperature to Kelvin

The temperature needs to be converted to Kelvin. This can be done by adding 273.15 to the Celsius temperature: Temperature in Kelvin = 27 + 273.15 = 300.15 K.

Use Ideal Gas Law

The Ideal Gas Law is given by PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the gas constant (0.0821 atm·L/mol·K), and T is the temperature in Kelvin. We need to solve for n (number of moles): n = PV/(RT).

Calculate Moles of CO2

Plug the values into the Ideal Gas Law to calculate n: n = (5 atm)(10 L) / (0.0821 atm·L/mol·K)(300.15 K).

Solve for Moles of CO2

Calculate the moles of CO2: n = 50 / (0.0821 * 300.15) moles.

Convert Moles to Grams

The molar mass of CO2 is 44.01 g/mol. The mass (m) of CO2 can be calculated as m = n * Molar Mass: m = n * 44.01 g/mol.

Final Calculation

After substituting the moles from Step 4 into the equation from Step 5, we get the mass in grams. This is the weight of CO2 in the cylinder.

Key concepts

Moles of Gas

Understanding the concept of moles is crucial when dealing with gas laws and chemical reactions. A mole is a unit used to express amounts of a chemical substance, defined as the amount of any substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12. This number is known as Avogadro's number, and it is approximately equal to \(6.022 \times 10^{23}\) entities per mole.

In terms of gases, a mole can be visualized as a specific volume that one mole of any gas occupies under certain conditions of temperature and pressure. According to the Ideal Gas Law, at standard temperature and pressure (STP), one mole of any gas occupies 22.4 liters. However, conditions often vary, so the Ideal Gas Law is used to determine the number of moles present in a given volume of gas at any temperature and pressure. The calculation involves rearranging the Ideal Gas Law equation to solve for \(n\), which denotes moles. Thus, in our example with the CO2 cylinder, the moles of CO2 were found by dividing the product of pressure and volume by the product of the gas constant and the temperature in Kelvin.

Gas Constant (R)

The Gas Constant, denoted as \(R\), is a fundamental component in the equation of the Ideal Gas Law, which relates pressure, volume, temperature, and the number of moles of gas in a system. Its value is constant and universally accepted as \(0.0821 \text{atm} \cdot \text{L/mol} \cdot \text{K}\), or equivalently, \(8.314 \text{J/mol} \cdot \text{K}\) when dealing with energy units. This constant is instrumental in ensuring that the Ideal Gas Law provides accurate results regardless of the units used for pressure, volume, and temperature.

When solving problems involving the Ideal Gas Law, it's essential to match the units of the gas constant to those of the other variables in the equation. If pressure is measured in atmospheres and volume in liters, then the version of \(R\) with atmospheres and liters must be used, as is the case in our example problem.

Molar Mass of CO2

Molar mass, another key concept in chemistry, is the mass of one mole of a substance, expressed in grams per mole (g/mol). It is crucial for converting between the mass of a substance and the number of moles. The molar mass of carbon dioxide (CO2) is determined by adding the atomic masses of one carbon atom (C) and two oxygen atoms (O).

Since carbon has an atomic mass of about 12.01 g/mol and oxygen has an atomic mass of about 16.00 g/mol, the molar mass of CO2 can be calculated as follows: \[ \text{Molar Mass of CO2} = 12.01 \text{g/mol} + (2 \times 16.00 \text{g/mol}) = 44.01 \text{g/mol} \]. Knowing the molar mass of CO2 allows us to convert from moles to grams, which is precisely what was needed for finding the mass of CO2 within the gas cylinder.

Temperature Conversion Kelvin

Temperature conversion is a fundamental step in solving many chemistry problems, especially when dealing with gas laws. The Kelvin scale is an absolute temperature scale where zero Kelvin (0 K) is theoretically the lowest possible temperature, defined as absolute zero.

For the Ideal Gas Law to work correctly, temperature must be in Kelvin. To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature. In the context of our example problem, the temperature is given in Celsius as \(27^\circ C\). Converted to Kelvin, the temperature becomes \(300.15 K\), which is used in subsequent calculations involving the Ideal Gas Law. Remember, the Kelvin and Celsius scales have the same size degree, so this conversion doesn't involve complex calculations, just a simple addition.