Question

Suppose two 200.0-L tanks are to be filled separately with the gases helium and hydrogen. What mass of each gas is needed to produce a pressure of \(2.70 \mathrm{~atm}\) in its respective tank at \(24^{\circ} \mathrm{C} ?\)

Short answer

To fill a 200.0-L tank with helium to 2.70 atm at 24°C, approximately 88.4 g of helium is needed. For a 200.0-L tank filled with hydrogen under the same conditions, approximately 44.2 g of hydrogen is required.

Step-by-step solution

Convert temperature to Kelvin

The temperature is given in Celsius (24°C), and we need to convert it to Kelvin. To do this, we simply add 273 to the Celsius value: \(T(K) = 24 + 273 = 297 K\)

Rearrange the ideal gas law formula for n (moles)

We need to find the number of moles of each gas needed to fill the tanks. So, let's rearrange the ideal gas law formula to solve for the number of moles (n): \(n = \frac{P \cdot V}{R \cdot T}\)

Calculate the moles of each gas required

Now, we can plug the values into the equation for helium: \(n_{He} = \frac{2.70\,\text{atm} \cdot 200.0\,\text{L}}{0.0821 \frac{L \cdot atm}{mol \cdot K} \cdot 297\,\text{K}} \approx 22.1\, \text{moles}\) And for hydrogen: \(n_{H2} = \frac{2.70\,\text{atm} \cdot 200.0\,\mathrm{L}}{0.0821 \frac{L \cdot atm}{mol \cdot K} \cdot 297\,\mathrm{K}} \approx 22.1\, \text{moles}\) The number of moles of each gas needed to fill their respective tanks to the desired pressure and temperature is approximately 22.1 moles.

Calculate the mass of each gas required

Finally, we'll use the molar mass (MM) of each gas to convert the number of moles needed (n) to mass (m). The mass (m) can be found with the following equation: \(m = n \cdot \mathrm{MM}\) For helium: \(m_{He} = 22.1\, \text{moles} \cdot 4.00\, \frac{\text{g}}{\mathrm{mol}} \approx 88.4\, \mathrm{g}\) For hydrogen: \(m_{H2} = 22.1\, \text{moles} \cdot 2.00\, \frac{\text{g}}{\mathrm{mol}} \approx 44.2\, \mathrm{g}\)

State the mass of each gas required to reach the desired conditions

The mass of helium needed to fill its tank to 2.70 atm at 24°C is approximately 88.4 g, and the mass of hydrogen needed to fill its tank to 2.70 atm at 24°C is approximately 44.2 g.

Key concepts

Molar Mass

Molar mass is an important concept in chemistry that allows us to link the mass of a substance to the amount of substance, measured in moles. It is commonly expressed in grams per mole (g/mol). For example, the molar mass of helium (He) is 4.00 g/mol, meaning one mole of helium weighs 4.00 grams. Similarly, hydrogen (H₂) has a molar mass of 2.00 g/mol.

Understanding molar mass is crucial when you want to convert between the mass of a substance and the number of moles. This is especially helpful in exercises involving gases, such as predicting how much of a gas is needed to achieve a certain pressure, as shown in our exercise. By using the molar mass along with the number of moles calculated from a gas law, you can find the precise mass required.

Gas Pressure

Gas pressure is the force that the gas exerts on the walls of its container. It is commonly measured in units like atmospheres (atm), Pascals, or millimeters of mercury (mmHg). In the context of the ideal gas law, gas pressure is essential because it influences how molecules of gas behave in relation to volume, temperature, and the number of moles.

For the given exercise, the tanks are intended to maintain a gas pressure of 2.70 atm. By using the ideal gas law, you can predict how changes in volume, temperature, or the amount of gas (measured in moles) can affect the pressure within a container. This understanding allows you to manipulate the variables to achieve the desired pressure.

Temperature Conversion

In chemistry, temperatures are frequently measured in Celsius, but many scientific calculations require Kelvin. Kelvin is the SI unit for temperature and is related directly to the concept of absolute temperature, starting at absolute zero. Converting from Celsius to Kelvin is simple: merely add 273 to your Celsius temperature.

For instance, in this exercise, the temperature given is 24°C. We convert this into Kelvin by calculating: \[ T(K) = 24 + 273 = 297 \text{ K} \] Making errors in temperature conversion can lead to incorrect outcomes in your calculations, especially when using the ideal gas law. Therefore, always ensure your temperatures are in Kelvin for these calculations.

Moles Calculation

Calculating moles is central to using the ideal gas law, which is expressed as \[ PV = nRT \] where \( n \) is the number of moles. Rearranging this equation to solve for \( n \), you have \[ n = \frac{PV}{RT} \] This formula is used to predict how many moles of a gas are present in a container under specific conditions of pressure, volume, and temperature. For the exercise, plugging in the values for both helium and hydrogen with this formula: \[ n = \frac{2.70 \times 200.0}{0.0821 \times 297} \approx 22.1 \text{ moles} \] Thus, you can determine how much gas is needed to produce the desired conditions. Having calculated the moles, you can then convert this into mass using molar mass, closing the loop between different measurements of gas properties.